Inequalities

Volume 9, Number 1 (2015), 257–276 doi:10.7153/jmi-09-23

## PARABOLIC FRACTIONAL MAXIMAL AND INTEGRAL OPERATORS WITH ROUGH KERNELS IN PARABOLIC GENERALIZED MORREY SPACES

VAGIFS. GULIYEV ANDAYDINS. BALAKISHIYEV

(*Communicated by S. Samko*)

*Abstract.* Let*P*be a real*n×n*matrix, whose all the eigenvalues have positive real part,*A**t*=*t*^{P},
*t**>*0 , γ=tr*P*is the homogeneous dimension on R^{n} andΩis an*A**t*-homogeneous of degree
zero function, integrable to a power *s**>*1 on the unit sphere generated by the corresponding
parabolic metric. We study the parabolic fractional maximal and integral operators *M*_{Ω,}^{P}_{α} and
*I*_{Ω,}^{P}_{α}, 0*<*α*<*γ with rough kernels in the parabolic generalized Morrey space *M**p,*ϕ*,P*(R^{n}).
Wefind conditions on the pair(ϕ1*,*ϕ2)for the boundedness of the operators *M*^{P}_{Ω,}_{α} and *I*_{Ω,}^{P}_{α}
from the space*M**p,*ϕ1*,P*(R^{n})to another one*M**q,*ϕ2*,P*(R^{n}), 1*<**p**<**q**<*∞, 1*/**p**−*1*/**q*=α*/*γ^{,}
and from the space *M*1*,*ϕ1*,**P*(R^{n})to the weak space *WM**q**,*ϕ2*,**P*(R^{n}), 1*q**<*∞, 1*−*1*/q*=
α*/*γ. We alsofind conditions onϕ for the validity of the Adams type theorems *M*_{Ω,}^{P}_{α}*,**I*_{Ω,}^{P}_{α}:
*M**p**,*ϕ^{1}^{p}*,**P*(R^{n})*→**M*

*q**,*ϕ^{1}^{q}*,**P*(R^{n})*,*1*<**p**<**q**<*∞.

## 1. Introduction

The boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operators, fractional integral operators and singular inte- gral operators etc, from one weighted Lebesgue space to another one is well studied by now, and there are well known various applications of such results in partial differential equations. Besides Lebesgue spaces, Morrey spaces, both the classical ones (the idea of their definition having appeared in [22]) and generalized ones, also play an important role in the theory of partial differential equations, see [12,19,20,29,30].

In this paper, wefind conditions for the boundedness of the parabolic fractional maximal and integral operators with rough kernel from a parabolic generalized Morrey space to another one, including also the case of weak boundedness, and prove Adams type boundedness theorems for this operators. To precisely formulate the results of this paper, we need the notions given below.

Note that we deal not exactly with the parabolic metric, but with a general aniso- tropic metric ρ of generalized homogeneity, the parabolic metric being its particular

*Mathematics subject classification*(2010): 42B20, 42B25, 42B35.

*Keywords and phrases*: Parabolic fractional maximal function, parabolic fractional integral, parabolic
generalized Morrey space.

The research of V. Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.003) and (PYO.FEN.4001.13.012).

c ^{} , Zagreb

Paper JMI-09-23 257

case, but we keep the term parabolic in the title and text of the paper, following the existing tradition, see for instance [7].

Everywhere in the sequel *AB* means that*ACB* with some positive constant
*C*independent of appropriate quantities. If*AB*and*BA*, we write*A≈B*and say
that*A* and*B* are equivalent.

**1.1. Parabolic homogeneous space***{*R^{n}*,*ρ*,dx}*

For*x∈*R^{n}and*r>*0, we denote the open ball centered at*x*of radius*r*by*B*(*x,r*),
its complement by ^{}*B*(*x,r*)and*|B*(*x,r*)*|*will stand for the Lebesgue measure of*B*(*x,r*).
Let*P*be a real*n×n*matrix, whose all the eigenvalues have positive real part. Let
*A*_{t}=*t*^{P} (*t>*0), and set

γ=tr*P.*

Then, there exists a quasi-distanceρ associated with*P*such that (see [8])
(*a*) ρ(*A*_{t}*x*) =*t*ρ(*x*)*,* *t>*0*,* for every *x∈*R^{n};

(*b*) ρ(0) =0*,* ρ(*x*) =ρ(*−x*)0

and ρ(*x−y*)*k*(ρ(*x−z*) +ρ(*y−z*));

(*c*) *dx*=ρ^{γ−1}^{d}σ(*w*)*d*ρ*,* where ρ=ρ(*x*)*,w*=*A*_{ρ}*−*1*x*

and*d*σ(*w*)is a measure on the unit ellipsoid *S*_{ρ}=*{w*:ρ(*w*) =1*}.*

Then, *{*R^{n}*,*ρ*,dx}* becomes a space of homogeneous type in the sense of Coifman-
Weiss (see [8]) and a homogeneous group in the sense of Folland-Stein (see [10]).

Moreover, we always assume that there hold the following properties of the quasidis-
tanceρ^{:}

(d) For every *x*,

*c*_{1}*|x|*^{α}^{1}ρ(*x*)*c*_{2}*|x|*^{α}^{2}*,* if ρ(*x*)1;

*c*_{3}*|x|*^{α}^{3}ρ(*x*)*c*_{4}*|x|*^{α}^{4}*,* if ρ(*x*)1
and

ρ(θ^{x})ρ(*x*) for 0*<*θ*<*1*,*

with some positive constantsα*i* and*c**i* (*i*=1*,...,*4). Similar properties hold also for
the quasimetricρ^{∗}associated with the adjoint matrix*P*^{∗}.

The following are some important examples of the above defined matrices*P* and
distancesρ^{.}

1. Let(*Px,x*)(*x,x*),*x∈*R^{n}. In this case,ρ(*x*)is defined as the unique solution
ρ(*x*) =*t* of*|A*_{t}*−*1*x|*=1, and*k*=1. This is the case studied by Calderon and Torchinsky
in [7].

2. Let*P*be a diagonal matrix with positive diagonal entries, and*t*=ρ(*x*),*x∈*R^{n}
be the unique solution of*|A*_{t}*−*1*x|*=1.

2_{a})When all the diagonal entries are greater than or equal to 1, O. V. Besov, V.

P. Il’in, P. I. Lizorkin in [3] and E. B. Fabes and N. M. Rivi`ere in [9] studied the weak
(1*,*1)and strong(*p,p*)estimates of singular integral operators.

2_{b})If there are diagonal entries smaller than 1, then ρ satisfies the above(*a*)*−*
(*d*)with*k>*1.

3. In [28] Stein and Wainger defined and studied some problems in harmonic
analysis on this kind of spaces. Consider a one parameter group of dilations onR^{n},
*A*_{t}:R^{n}*→*R^{n} for each*t>*0 with the following properties:

(i) *A*_{st}=*A*_{s}*A*_{t} and *A*_{1} is the identity;

(ii) lim

*t→*0*A*_{t}*x*=0 for every *x∈*R^{n};
(iii) *A*_{t}*x*=*t*^{P}*x*=exp*{P*log*t}x*.

Then all eigenvalues of *P* have positive real part. Then in this case there exist
0*<*β1*<*β2 and 0*<c*^{}_{1}*<c*^{}_{2} such that*A*_{t} has the following properties:

(iv) for every*x*

*c*^{}_{1}*t*^{β}^{1}*|x|<|A**t**x|<c*^{}_{2}*t*^{β}^{2}*|x|,* for*t*1
and

(*c*^{}_{2})^{−1}*t*^{β}^{2}*|x|<|A**t**x|<*(*c*^{}_{1})^{−1}*t*^{β}^{1}*|x|,* for *t*1*.*

By [28], if*|A*_{tx}*|*were strictly monotonic, then we might define the unique solutions
of*|A**t**x|*=1 byρ(*x*). Otherwise, there is a positive definite symmetric matrix*B* such
that

*A**t**x*=*A**t**x*_{B}= (*BA**t**x,A**t**x*)^{1}^{2}

is strictly increasing and thus ρ can be defined as follows: For *x *=0, ρ(*x*) is the
unique positive*t* such that*A**t*^{−1}(*x*)=1. For*x*=0, setρ(*x*) =0. Then for *x *=0

*x*=*A*_{ρ(x)}*w*(*x*)*,*

where*w*(*x*)=1 and*w*(*x*)is unique. Let ρ^{∗}(ξ)be the quasi-distance function corre-
sponding to the group*A*_{t}^{∗}=*t*^{P}^{∗}=exp(*P*^{∗}log*t*). Thenξ =*A*^{∗}_{ρ}_{∗}_{(ξ)}(*w*^{∗}(ξ))where

*w*^{∗}(ξ)= (*B*1*w*^{∗}(ξ)*,w*^{∗}(ξ))^{1}^{2}
for an appropriate positive definite symmetric matrix *B*.

It was pointed out in [28] that both ρ ^{and} ρ^{∗} satisfy (1.1)–(1.4), and one can
easily see thatα1=α4=_{β}^{1}_{2},α1=α3=_{β}^{1}_{1}, and*c**i*depends on on the matrices *P*and
*B*. Moreover, in this case

*dx*=ρ^{γ−1}^{d}σ(*w*)*d*ρ*,*

where*d*σ(*w*)is a*C*^{∞} measure on the ellipsoidρ(*w*)^{2}= (*Bw,w*) =1.

In the standard parabolic case*P*_{0}=diag(1*,...,*1*,*2)we have

ρ(*x*) =

*|x*^{}*|*^{2}+

*|x*^{}*|*^{4}+*x*^{2}_{n}

2 *,* *x*= (*x*^{}*,x*_{n})*.*

The balls *E*(*x,r*) =*{y∈*R^{n}:ρ(*x−y*)*<r}* with respect to the quasidistance ρ
are ellipsoids. For its Lebesgue measure one has

*|E*(*x,r*)*|*=*v*_{ρ}*r*^{γ}*,*

where*v*_{ρ} is the volume of the unit ellipsoid. By ^{}*E*(*x,r*) =R^{n}*\E*(*x,r*)we denote the
complement of*E*(*x,r*).

## 1.2. Parabolic generalized Morrey spaces

We define the parabolic Morrey space*M*_{p,λ,P}(R^{n})via the norm

*f*_{M}_{p,}_{λ}_{,P} = sup

*x∈*R^{n}*,t>*0

*t*^{−λ}

*E*(*x,t*)*|f*(*y*)*|*^{p}*dy*
_{1/p}

*<*∞*,*
where 1*p*∞and 0λ γ^{.}

If λ =0, then *M**p,*0*,P*(R^{n}) =*L*_{p}(R^{n}); if λ =γ^{, then} *M*_{p,γ,P}(R^{n}) =*L*_{∞}(R^{n}); if
λ*<*0 or λ*>*γ^{, then}*M*_{p,λ}_{,P}=Θ, whereΘ is the set of all functions equivalent to 0
onR^{n}.

We also denote by *WM*_{p,λ,P}(R^{n}) the weak parabolic Morrey space of functions
*f∈W L*^{loc}_{p} (R^{n})for which

*f*_{W}_{M}_{p}_{,}_{λ}_{,}_{P}= sup

*x∈*R^{n}*,t>*0*r*^{−}^{λ}^{p}*f**W L**p*(*E*(*x,r*))*<*∞*,*

where*W L*_{p}(*E*(*x,r*))denotes the weak *L*_{p}-space of measurable functions *f* for which
*f*_{WL}_{p}_{(E}_{(x,r))}=sup

*t>*0*t|{y∈E*(*x,r*):*|f*(*y*)*|>t}|*^{1/p}*.*
Note that*W L**p*(R^{n}) =*WM**p,*0*,P*(R^{n}),

*M*_{p,λ}_{,P}(R^{n})*⊂WM*_{p,λ}_{,P}(R^{n}) and *f*_{W}_{M}_{p,}_{λ}_{,P}*f*_{M}_{p,}_{λ}_{,P}*.*
If *P*=*I*, then *M*_{p,λ}(R^{n})*≡M*_{p,λ}_{,I}(R^{n})is the classical Morrey space.

We introduce the parabolic generalized Morrey spaces following the known ideas of defining generalized Morrey spaces ([15,24,26] etc).

DEFINITION1.1. Let ϕ(*x,r*) be a positive measurable function on R^{n}*×*(0*,*∞)
and 1*p<*∞. The space *M*_{p,ϕ,P}*≡M*_{p,ϕ,P}(R^{n})*,* called the parabolic generalized
Morrey space, is defined by the norm

*f**M*_{p,}_{ϕ}_{,P} = sup

*x∈*R^{n}*,t>*0ϕ(*x,t*)^{−1}*|E*(*x,t*)*|*^{−}^{1}^{p}*f*_{L}_{p}_{(E}_{(x,t))}*.*

DEFINITION1.2. Let ϕ(*x,r*) be a positive measurable function on R^{n}*×*(0*,*∞)
and 1*p<*∞. The space *WM*_{p,ϕ,P}*≡WM*_{p,ϕ,P}(R^{n})*,* called the weak parabolic
generalized Morrey space, is defined by the norm

*f**WM*_{p,}_{ϕ}_{,P} = sup

*x∈*R^{n}*,t>*0ϕ(*x,t*)^{−1}*|E*(*x,t*)*|*^{−}^{1}^{p}*f**W L**p*(*E*(*x,t*))*.*

If*P*=*I*, then*M*_{p,ϕ}(R^{n})*≡M*_{p,ϕ,I}(R^{n})and*WM*_{p,ϕ}(R^{n})*≡WM*_{p,ϕ,I}(R^{n})are the
generalized Morrey space and the weak generalized Morrey space, respectively.

According to this definition, we recover the space *M*_{p,λ}_{,P}(R^{n}) under the choice
ϕ(*x,r*) =*r*^{λ}^{−}^{p}^{γ}:

*M*_{p,λ}_{,P}(R^{n}) =*M*_{p,ϕ,P}(R^{n})

ϕ(*x,r*)=*r*^{λ}^{−}^{p}^{γ}

*.*

## 1.3. Operators under consideration

Let *S*_{ρ} =*{w∈*R^{n}:ρ(*w*) =1*}* be the unit ρ-sphere (ellipsoid) in R^{n} (*n*2)
equipped with the normalized Lebesgue surface measure *d*σ ^{and} Ω be *A*_{t}-homoge-
neous of degree zero, i.e. Ω(*A*_{t}*x*)*≡*Ω(*x*), *x∈*R^{n}, *t* *>*0. The parabolic fractional
maximal function *M*_{Ω,α}^{P} *f* and the parabolic fractional integral *I*_{Ω,α}^{P} *f* by with rough
kernels, 0*<*α*<*γ*,*of a function *f∈L*^{loc}_{1} (R^{n})are defined by

*M*_{Ω,α}^{P} *f*(*x*) =sup

*t>*0*|E*(*x,t*)*|*^{−1+}^{α}^{γ}^{}

*E*(*x,t*)*|*Ω(*x−y*)*| |f*(*y*)*|dy,*

*I*_{Ω,α}^{P} *f*(*x*) =^{}

R^{n}

Ω(*x−y*) *f*(*y*)
ρ(*x−y*)^{γ−α} *dy.*

If Ω*≡*1, then *M*_{α}^{P}*≡M*_{1,α}^{P} and *I*_{α}^{P}*≡I*_{1,α}^{P} are the parabolic fractional maximal
operator and the parabolic fractional integral operator, respectively. If α =0, then
*M*_{Ω}^{P}*≡M*_{Ω,0}^{P} is the parabolic maximal operator with rough kernel. If*P*=*I*, then*M*_{Ω,α}*≡*
*M*_{Ω,α}^{I} is the fractional maximal operator with rough kernel, and*M≡M*_{Ω,0}^{I} is the Hardy-
Littlewood maximal operator with rough kernel. It is well known that the parabolic
fractional maximal operators play an important role in harmonic analysis (see [10,27]).

We prove the boundedness of the parabolic fractional maximal and integral op-
erators *M*_{Ω,α}^{P} , *I*_{Ω,α}^{P} with rough kernel from one parabolic generalized Morrey space
*M*_{p,ϕ}1*,P*(R^{n}) to another one *M*_{q,ϕ}2*,P*(R^{n}), 1*<p<q<*∞, 1*/p−*1*/q*=α*/*γ^{, and}
from the space*M*_{1,ϕ}1*,P*(R^{n})to the weak space*WM*_{q,ϕ}2*,P*(R^{n}), 1*q<*∞, 1*−*1*/q*=
α*/*γ. We also prove the Adams type boundedness of the operators *M*_{Ω,α}^{P} , *I*_{Ω,α}^{P} from
*M**p,*ϕ^{1}^{p}*,P*(R^{n}) to *M*

*q,*ϕ^{1}^{q}*,P*(R^{n}) for 1 *<* *p* *<* *q* *<* ∞ and from *M*_{1,ϕ,P}(R^{n}) to
*WM*

*q,*ϕ^{1}^{q}*,P*(R^{n})for 1*<q<*∞.

## 2. Preliminaries

In the papers [24, 25], where the maximal and other operator were studied in
generalized Morrey spaces, the following condition was imposed onϕ(*x,r*):

*c*^{−1}ϕ(*x,r*)ϕ(*x,t*)*c*ϕ(*x,r*)*,* (2.1)
whenever*rt*2*r*, jointly with the condition:

_{∞}

*r* ϕ(*x,t*)^{p}*dt*

*t* *C*ϕ(*x,r*)^{p}
for the maximal or singular operators and the condition

_{∞}

*r* *t*^{α}^{p}ϕ(*x,t*)^{p}*dt*

*t* *C r*^{α}^{p}ϕ(*x,r*)^{p} (2.2)
for potential and fractional maximal operators, where*c*and*C*do not depend on *r*and
*x*.

The results of [24,25] imply the following statement.

THEOREM2.1. *Let* 1*p* *<*∞*,* 0*<*α *<* ^{γ}_{p}*,* ^{1}_{q} = ^{1}_{p}*−*^{α}_{γ} *and* ϕ(*x,t*) *satisfy*
*the conditions*(2.1)*and* (2.2)*. Then M*_{α}^{P} *and I*_{α}^{P} *are bounded from* *M*_{p,ϕ,P}(R^{n}) *to*
*M*_{q,ϕ,P}(R^{n})*for p>*1 *and fromM*_{1,ϕ,P}(R^{n})*to WM*_{q,ϕ,P}(R^{n})*for p*=1*.*

In [11] the following statement was proved by fractional integral operator with
rough kernels*I*_{Ω,α}, containing the result in [21,24].

THEOREM2.2. *Let*Ω*∈L*_{s}(*S*^{n−1})*,* 1*<s*∞*, be homogeneous of degree zero.*

*Let*0*<*α*<n, s*^{}*<p<*_{α}^{n}*,* ^{1}_{q}=^{1}_{p}*−*^{α}_{n} *and*ϕ(*x,r*)*satisfy the condition*(2.1)*and*

_{∞}

*r* *t*^{αp}ϕ(*x,t*)^{p}*dt*

*t* *C r*^{α}^{p}ϕ(*x,r*)^{p}*,* (2.3)
*where C does not depend on x and r . Then the operator I*_{Ω,α} *is bounded from M*_{p,ϕ}
*to M*_{q,ϕ}*.*

The following statements, containing results obtained in [21], [24] was proved in [13,15] (see also [4]–[6], [14]–[17]).

THEOREM2.3. *Let*0*<*α*<*γ^{,}^{1}*p<*_{α}^{γ}*,* ^{1}_{q}= ^{1}_{p}*−*^{α}_{γ} *and*(ϕ1*,*ϕ2)*satisfy the*

*condition* _{}

∞

*r* *t*^{α−1}ϕ1(*x,t*)*dtC*ϕ2(*x,r*)*,* (2.4)
*where C does not depend on x and r . Then the operator I*_{α}^{P} *is bounded from M*_{p,ϕ}_{1}_{,P}
*to M*_{q,ϕ}_{2}_{,P} *for p>*1 *and from M*_{1,ϕ}_{1}_{,P}*to W LM*_{q,ϕ}_{2} *for p*=1*.*

Let *v* be a weight on (0*,*∞)*.* We denote by *L*_{∞,v}(0*,*∞) the space of all functions
*g*(*t*),*t>*0 withfinite norm

*g**L*_{∞,v}(0*,*∞)=ess sup

*t>*0 *v*(*t*)*|g*(*t*)*|*

and write*L*_{∞}(0*,*∞)*≡L*_{∞,1}(0*,*∞). Let M(0*,*∞) be the set of all Lebesgue-measurable
functions on (0*,*∞) and M^{+}(0*,*∞) its subset of all nonnegative functions. By
M^{+}(0*,*∞;*↑*)we denote the cone of all functions inM^{+}(0*,*∞)non-decreasing on(0*,*∞)
and introduce also the set

A=

ϕ*∈*M^{+}(0*,*∞;*↑*): lim

*t→*0+ϕ(*t*) =0 *.*

Let*u*be a non-negative continuous function on (0*,*∞). We define the supremal opera-
tor*S**u*on *g∈*M(0*,*∞)by

(*S*_{u}*g*)(*t*):=*u g**L*_{∞}(*t,*∞)*,* *t∈*(0*,*∞)*.*

The following theorem was proved in [5].

THEOREM2.4. *Let v*_{1}*, v*_{2} *be non-negative measurable functions satisfying* 0*<*

*v*1_{L}_{∞}_{(t,∞)} *<*∞ *for any t>*0 *and let u be a continuous non-negative function on*
(0*,*∞)*.Then the operator S*_{u}*is bounded from L*_{∞,v}_{1}(0*,*∞)*to L*_{∞,v}_{2}(0*,*∞)*on the cone*A
*if and only if*

*v*_{2}*S**u*

*v*1^{−1}_{L}_{∞}_{(·,∞)}

*L*_{∞}(0*,*∞)*<*∞*.* (2.5)

We are going to use the following statement on the boundedness of the weighted Hardy operator

*H*_{w}^{∗}*g*(*t*):=^{} ^{∞}

*t* *g*(*s*)*w*(*s*)*ds,* 0*<t<*∞*,*
where*w*is afixed function non-negative and measurable on(0*,*∞).

The following theorem was proved in [18].

THEOREM2.5. *Let v*_{1}*, v*_{2}*and w be positive almost everywhere and measurable*
*functions on*(0*,*∞)*. The inequality*

ess sup

*t>*0 *v*_{2}(*t*)*H*_{w}^{∗}*g*(*t*)*C*ess sup

*t>*0 *v*_{1}(*t*)*g*(*t*) (2.6)

*holds for some C>*0 *for all non-negative and non-decreasing g on*(0*,*∞) *if and only*
*if*

*B*:=ess sup

*t>*0 *v*_{2}(*t*)

_{∞}

*t*

*w*(*s*)*ds*
ess sup

*s<*τ*<*∞*v*1(τ)*<*∞*.* (2.7)
*Moreover, if C*^{∗}*is the minimal value of C in*(2.6)*, then C*^{∗}=*B.*

REMARK2.1. In (2.6) and (2.7) it is assumed that _{∞}^{1} =0 and 0*·*∞=0.

**3. Boundedness of the parabolic fractional**
**operators in the spaces***L**p*(R^{n})

In this section we prove the (*p,p*)-boundedness of the operator *M*_{Ω}^{P} and the
(*p,q*)-boundedness of the operators*I*_{Ω,α}^{P} and*M*_{Ω,α}^{P} .

THEOREM3.1. *Let*Ω*∈L*_{s}(*S*_{ρ})*,*1*<s*∞*, be A*_{t}*-homogeneous of degree zero.*

*Then the operator M*_{Ω}^{P} *is bounded in the space L*_{p}(R^{n})*, p>s*^{}*.*

*Proof.* In the case*s*=∞the statement of Theorem3.1is known and may be found
in [8] and [27]. So we assume that 1*<s<*∞.

Note that

Ω(*x− ·*)*L**s*(*E*(*x,t*))=^{}

*E*(0*,t*)*|*Ω(*y*)*|*^{s}*dy*_{1/s}

=^{} *t*
0*r*^{γ−1}*dr*

*S*_{ρ}*|*Ω(ω)*|*^{s}*d*σ(ω)_{1/s}

(3.1)

=*c*_{0}Ω*L**s*(*S*ρ)*|E*(*x,t*)*|*^{1/s}*,*
where*c*_{0}=

γ^{v}ρ_{−1/s}

and*v*_{ρ}=*|E*(0*,*1)*|*.

The case *p*=∞is easy. Indeed, making use of (3.1), we get
*M*_{Ω}^{P}*f**L*_{∞}*f**L*_{∞}sup

*t>*0*|E*(*x,t*)*|*^{−1+}^{s}^{1}Ω(*x− ·*)_{L}_{s}_{(E}_{(x,t))}*c*_{0}Ω_{L}_{s}_{(S}_{ρ}_{)}*f**L*_{∞}*.*
So we assume that*s*^{}*<p<*∞*.*Applying H¨older’s inequality, we get

*M*_{Ω}^{P}*f*(*x*)sup

*t>*0*|E*(*x,t*)*|*^{−1}Ω(*x− ·*)*L**s*(*E*(*x,t*))*f**L*_{s}(*E*(*x,t*))*.* (3.2)
Then from (3.2) and (3.1) we have

*M*_{Ω}^{P}*f*(*x*)*c*0Ω*L**s*(*S*ρ)sup

*t>*0*|E*(*x,t*)*|*^{−1+1/s}*f**L*_{s}(*E*(*x,t*))

=*c*0Ω*L**s*(*S*ρ)

sup

*t>*0*|E*(*x,t*)*|*^{−1}*|f|*^{s}^{}*L*1(*E*(*x,t*))

_{1/s}

=*c*_{0}Ω*L**s*(*S*ρ)

*M*^{P}(*|f|*^{s}^{})(*x*)_{1/s}

*.* (3.3)

Therefore, from (3.3) for 1*s*^{}*<p<*∞we get
*M*_{Ω}^{P}*f**L**p**c*_{0}Ω_{L}_{s}_{(S}_{ρ}_{)}

*M*^{P}(*|f|*^{s}^{})(*x*)_{1/s}^{}
*L**p*

=*c*_{0}Ω*L**s*(*S*ρ)*M*^{P}*|f|*^{s}^{}^{1/s}_{L}_{p}^{}

*/**s* *|f|*^{s}^{}^{1/s}_{L}_{p}^{}

*/**s* =*f**L**p**.*

THEOREM3.2. *Suppose that* 0*<*α*<*γ *and the function* Ω*∈L* γ

γ*−*α(*S*_{ρ})*, is A**t**-*
*homogeneous of degree zero. Let*1*p<*_{α}^{γ} *and*1*/p−*1*/q*=α*/*γ*. Then the fractional*
*integral operator I*_{Ω,α}^{P} *is bounded from L*_{p}(R^{n})*to L*_{q}(R^{n})*for p>*1*and from L*_{1}(R^{n})
*to W L*_{q}(R^{n})*for p*=1*.*

*Proof.* We denote

*K*(*x*):= Ω(*x*)
ρ(*x*)^{γ−α}
for brevity, and may assume that*K*(*x*)0. We have

*{x∈*R^{n}:*I*_{α}^{P}*f*(*x*)*>*λ*}{x∈*R^{n}:*I*_{Ω}^{P}*f*(*x*)*>C*_{γ,α}^{−1}λ*}I*_{1}+*I*_{2}*,*

where
*I*_{1}:=

*x∈*R^{n}:*|K*^{1}_{μ}*∗f*(*x*)*|>*λ

2*,* *I*_{2}:=

*x∈*R^{n}:*|K*_{μ}^{2}*∗f*(*x*)*|>*λ
2*,*
*K*_{μ}^{1}(*x*) = (*K*(*x*)*−*μ)χ*E*(μ)(*x*) and *K*_{μ}^{2}(*x*) =*K*(*x*)*−K*_{μ}^{1}(*x*)*,*
μ*>*0 and *E*(μ) =*{x∈*R^{n}:*K*(*x*)*>*μ*}*. Note that

*|E*(μ)*|B*μ^{γ}^{−}^{γ}^{α}*.* (3.4)

where*B*=_{α}^{1}Ω_{L}^{γ}^{−}^{γ}^{α}_{γ}

γ*−*α(*S*_{ρ}) as seen from the following estimation:

*|E*(μ)*|* 1
μ

*E*(μ)

*|*Ω(*x*)*|*

ρ(*x*)^{γ−α}*dx*

= 1 μ

*S*_{ρ}Ω(*x*^{})*d*σ(*x*^{})^{}
_{|Ω(}_{x}_{)|}

μ

_{γ}_{−}^{1}_{α}

0 *r*^{α−1}*dr*=*B*μ^{γ}^{−}^{γ}^{α}*.*
By means of (3.4) we can prove the estimate

*K*_{μ}^{2}*L*_{p} γ*−*α
γ ^{Bq}

_{p}^{1}

μ^{(}^{γ}^{−}^{γ}^{α}^{)}^{q}*,* 1*p<* γ
α^{.}
For*p*=1 it easily follows from (3.4), and for *p>*1 we have

R^{n}*|K*_{μ}^{2}(*x*)*|*^{p}^{}*dx*=*p*^{}

_{μ}

0 *t*^{p}^{}^{−1}*|E*(*t*)*|dt*
*p*^{}*B*^{} ^{μ}

0 *t*^{p}^{}^{−1−}^{γ}^{−}^{γ}^{α}*dt*

=γ*−*α

γ ^{Bq}μ^{γ}^{−}^{γ}^{α}^{p}^{q}^{}*.*

Then by the Young inequality we obtain
*K*_{μ}^{2}*∗f**L*_{∞}*K*_{μ}^{2}*L*_{p}*f**L**p*γ*−*α

γ ^{Bq}
_{p}^{1}

μ^{(}^{γ}^{−}^{γ}^{α}^{)q} *f**L**p**.*
Now for a λ*>*0, we choose μ ^{such that}

γ*−*α
γ ^{Bq}

_{p}^{1}

μ^{(}^{γ}^{−}^{γ}^{α}^{)}^{q} *f**L**p* =λ
2*,*
then

*x∈*R^{n}:*|K*_{μ}^{2}*∗f*(*x*)*|>*λ
2=0*.*
Thus

*{x∈*R^{n}:*I*_{α}^{P}*f*(*x*)*>*λ*}*

*x∈*R^{n}:*|K*_{μ}^{1}*∗f*(*x*)*|>*λ
2

2

λ ^{K}^{μ}^{1}^{∗}^{f}^{}^{L}^{p}
_{p}

*.* (3.5)

The following estimations take (3.4) into account:

R^{n}*|K*_{μ}^{1}(*x*)*|dx*=^{}

*E*(μ)

*|K*(*x*)*| −*μ^{}^{dx}

^{} ^{∞}

0 *|E*(*t*+μ)*|dt*

*B*

_{∞}

μ *t*^{−}^{γ}^{−}^{γ}^{α}*dt* (3.6)

= α^{B}

γ*−*α μ^{−}^{γ}^{−}^{α}^{α}*.*
For all *f* *∈L*_{∞}(R^{n})and*x∈*R^{n}, from (3.6) it follows that

*|K*^{1}_{μ}*∗f*(*x*)*|f**L*_{∞}

R^{n}*|K*^{1}_{μ}(*x*)*|dx* α^{B}

γ*−*α μ^{−}^{γ}^{−}^{α}^{α} *f**L*_{∞}*.* (3.7)
For all *f* *∈L*_{1}(R^{n}), from (3.6) follows

*K*_{μ}^{1}*∗f**L*1^{}

R^{n}

R^{n}*|K*_{μ}^{1}(*x−y*)*||f*(*y*)*|dxdy* α^{B}

γ*−*α μ^{−}^{γ}^{−}^{α}^{α} *f**L*1*.* (3.8)
Thus from (3.7) and (3.8) follows that the operator*T*_{1}:*f→K*_{μ}^{1}*∗f* is of(∞*,*∞)and
(1*,*1)-type. Then by the Riesz-Thorin theorem the operator *T*_{1} is also of (*p,p*)-type,
1*<p<*∞, and

*T*_{1}*f**L**p* α^{B}

γ*−*α μ^{−}^{γ}^{−}^{α}^{α} *f**L**p**.* (3.9)

From (3.5) and (3.9) we get

*{x∈*R^{n}:*I*_{α}^{P}*f*(*x*)*>*λ*}*2

λ ^{}^{K}^{1}^{μ}^{∗}^{f}^{}^{L}^{p}
_{p}

*C*1

λ ^{}^{f}^{}^{L}^{p}
*q*

*,* (3.10)

where*C*is independent ofλ ^{and} ^{f}^{.}

To finish the proof, i.e. prove that the operator *I*_{α}^{P} is bounded from *L*_{p}(R^{n}) to
*L*_{q}(R^{n}) for 1*<p<*_{α}^{γ} and 1*/p−*1*/q*=α*/*γ, observe that the inequality (3.10) tells
us that*I*_{α}^{P} is bounded from*L*_{1}(R^{n})to*W L**q*(R^{n})with 1*−*1*/q*=α*/*γ. We choose any
*p*_{0} such that *p<p*_{0}*<*_{α}^{γ}, and put _{q}^{1}_{0} = _{p}^{1}_{0}*−*^{α}_{γ}. By (3.10) the operator*I*_{α}^{P} is of weak
(*p*0*,q*_{0})-type. Since it is also of weak(1*,q*)-type by the Marcinkiewicz interpolation
theorem, we conclude that*I*_{α}^{P} is of(*p,q*)-type.

COROLLARY3.1. *Under the assumptions of Theorem*3.2*, the fractional maximal*
*operator M*_{Ω,α}^{P} *is bounded from L**p*(R^{n}) *to L**q*(R^{n}) *for p>*1 *and from L*_{1}(R^{n}) *to*
*W L*_{q}(R^{n})*for p*=1*.*

*Proof.* It suffices to refer to the known fact that

*M*_{Ω,α}^{P} *f*(*x*)*C*_{γ,α}*I*_{Ω,α}^{P} *f*(*x*)*,* *C*_{γ,α}=*|E*(0*,*1)*|*^{γ}^{−}^{γ}^{α}*,*
Note that in the isotropic case *P*=*I* Theorem3.2was proved in [23].

**4. Parabolic fractional maximal operator with rough kernels in the spaces**
*M*_{p,ϕ,P}(R^{n})

Note that in the next Section 5 we obtain boundedness results of Spanne and
Adams type for the fractional integral operator *I*_{Ω,α}^{P} . Although *M*^{P}_{Ω,α}*f* is dominated
by*I*_{Ω,α}^{P} *f* and consequently from the results of Section5there may be derived the cor-
responding results for*M*_{Ω,α}^{P} *f*, we obtain here a Spanne type statement for the operator
*M*_{Ω,α}^{P} *f* separately from Section5, because for this operator we are able to obtain the
boundedness results under weaker assumptions than in Section5, see Remark5.3.

Recall that in the classical isotropic case, i.e. in the case of of the operator *M*_{α},
0α*<n*onR^{n}with Euclidean distance, sufficient conditions on for the boundedness
of this operator in generalized Morrey spaces*M**p,*ϕ(R^{n}) have been obtained in [2,5,
15,24].

LEMMA4.1. *Suppose that* 0*<*α *<*γ *and the function* Ω*∈L* γ

γ*−*α(*S*_{ρ}) *is A*_{t}*-*
*homogeneous of degree zero. Let*1*p<*_{α}^{γ}*,* ^{1}_{q}=^{1}_{p}*−*^{α}_{γ}*. Then for any ballE* =*E*(*x,r*)
*in*R^{n} *and f∈L*^{loc}_{p} (R^{n})*there hold the inequalities*

*M*_{Ω,α}^{P} *f**L**q*(*E*(*x,r*))*f**L**p*(*E*(*x,*2*kr*))+*r*^{γ}^{q} sup

*t>*2*kr**t*^{−γ+α}Ω(*x− ·*)*f*(*·*)*L*1(*E*(*x,t*))*,* *p>*1*,*

*M*_{Ω,α}^{P} *f**W L**q*(*E*(*x,r*))*f**L*1(*E*(*x,*2*kr*))+*r*^{γ}^{q} sup

*t>*2*kr**t*^{−γ+α}Ω(*x− ·*)*f*(*·*)*L*1(*E*(*x,t*))*,* *p*=1*.*
(4.1)
*Proof.* Given a ball *E* =*E*(*x,r*)*,* we split the function *f* as *f* = *f*_{1}+*f*_{2}, where
*f*_{1}=*f*χ*E*(*x,*2*kr*) and *f*_{2}=*f*χ(*E*(*x,*2*kr*))*,*and then

*M*^{P}_{Ω,α}*f**L**q*(*E*)*M*^{P}_{Ω,α}*f*1*L**q*(*E*)+*M*_{Ω,α}^{P} *f*2*L**q*(*E*)*.*
Let *p>*1*.*By Corollary3.1

*M*_{Ω,α}^{P} *f*_{1}*L**q*(*E*)*f**L**p*(*E*(*x,*2*kr*))*.*

To estimate*M*_{Ω,α}^{P} *f*_{2}(*y*)*,* observe that if*E*(*y,t*)*∩*^{}(*E*(*x,*2*kr*))* *=/0*,* where*y∈E*, then
*t>r*. Indeed, if *z∈E*(*y,t*)*∩*^{}(*E*(*x,*2*kr*))*,*then*t>*ρ(*y−z*)^{1}_{k}ρ(*x−z*)*−*ρ(*x−y*)*>*

2*r−r*=*r*.

On the other hand, *E*(*y,t*)*∩*^{}(*E*(*x,*2*kr*))*⊂E*(*x,*2*kt*). Indeed, for *z∈E*(*y,t*)*∩*

(*E*(*x,*2*kr*))we getρ(*x−z*)*k*ρ(*y−z*) +*k*ρ(*x−y*)*<k*(*t*+*r*)*<*2*kt*.

Hence

*M*_{Ω,α}^{P} *f*_{2}(*y*) =sup

*t>*0

1

*|E*(*y,t*)*|*^{1−α/γ}

*E*(*y,t*)*∩*^{}(*E*(*x,*2*kr*))*|f*(*z*)*||*Ω(*x−z*)*|dz*
(2*k*)^{γ−α}sup

*t>r*

1

*|E*(*x,*2*kt*)*|*^{1−α/γ}

*E*(*x,*2*kt*)*|f*(*z*)*||*Ω(*x−z*)*|dz*

= (2*k*)^{γ−α} sup

*t>*2*kr*

1

*|E*(*x,t*)*|*^{1−α/γ}

*E*(*x,t*)*|f*(*z*)*||*Ω(*x−z*)*|dz.*
Therefore, for all*y∈E* we have

*M*_{Ω,α}^{P} *f*_{2}(*y*)(2*k*)^{γ−α} sup

*t>*2*kr*

1

*|E*(*x,t*)*|*^{1−α/γ}

*E*(*x,t*)*|f*(*z*)*||*Ω(*x−z*)*|dz.* (4.2)
Thus

*M*_{Ω,α}^{P} *f**L**q*(*E*)*f**L**p*(*E*(*x,*2*kr*))+*|E|*^{1}^{q} sup

*t>*2*kr*

1

*|E*(*x,t*)*|*^{1−α/γ}

*E*(*x,t*)*|f*(*z*)*||*Ω(*x−z*)*|dz.*
Let *p*=1. We have

*M*_{Ω,α}^{P} *f**W L**q*(*E*)*M*_{Ω,α}^{P} *f*_{1}*WL**q*(*E*)+*M*_{Ω,α}^{P} *f*_{2}*W L**q*(*E*)*.*
By Corollary3.1we get

*M*_{Ω,α}^{P} *f*_{1}*WL**q*(*E*)*f**L*1(*E*(*x,*2*kr*))*.*
Then by (4.2) we arrive at (4.1) and complete the proof.

Similarly to Lemma4.1and Theorem3.1the following lemma may be proved.

LEMMA4.2. *Let the function* Ω*∈L**s*(*S*_{ρ})*,* 1*<s*∞*, be A**t**-homogeneous of*
*degree zero. Then for p>s*^{} *and any ballE* =*E*(*x,r*)*the inequality*

*M*_{Ω}^{P}*f*_{L}_{p}_{(E}_{(x,r))}*f*_{L}_{p}_{(E}_{(x,2kr))}+*r*^{γ}^{p} sup

*t>*2*kr**t*^{−γ}Ω(*x− ·*)*f*(*·*)_{L}_{1}_{(E}_{(x,t))}
*holds for all f∈L*^{loc}_{1} (R^{n})*.*

LEMMA4.3. *Suppose that the function*Ω*∈L* γ

γ*−*α(*S*_{ρ})*is A*_{t}*-homogeneous of de-*
*gree zero. Let*0*<*α*<*γ^{,} ^{1}*p<*_{α}^{γ}*,* ^{1}_{q}=^{1}_{p}*−*^{α}_{γ}*. Then for f* *∈L*^{loc}_{p} (R^{n}) *there hold*
*the inequalities*

*M*_{Ω,α}^{P} *f**L**q*(*E*(*x,r*))*r*^{γ}^{q} sup

*t>*2*kr**t*^{−}^{γ}^{q}*f**L**p*(*E*(*x,t*))*,* *p>*1*,* (4.3)
*M*_{Ω,α}^{P} *f**WL**q*(*E*(*x,r*))*r*^{γ}^{q} sup

*t>*2*kr**t*^{−}^{γ}^{q}*f**L*1(*E*(*x,t*))*,* *p*=1*.* (4.4)
*Proof.* Let *p>*1 Denote

*A*1:=*|E|*^{1}^{q}

*t>*2*kr*sup
1

*|E*(*x,t*)*|*^{1−α/γ}

*E*(*x,t*)*|f*(*z*)*||*Ω(*x−z*)*|dz*

*,*
*A*2:=*f**L**p*(*E*(*x,*2*kr*))*.*

Applying H¨older’s inequality, we get
*A*1*|E|*^{1}^{q} sup

*t>*2*kr**f**L**p*(*E*(*x,t*))Ω(*x− ·*)*L* γ

γ*−*α(*E*(*x,t*))*|E*(*x,t*)*|*^{α}^{γ}^{−}^{1}^{p}
*|E|*^{1}^{q} sup

*t>*2*kr**|E*(*x,t*)*|*^{−}^{1}^{q}*f**L**p*(*E*(*x,t*))*.*
On the other hand,

*|E|*^{1}^{q} sup

*t>*2*kr**|E*(*x,t*)*|*^{−}^{1}^{q}*f**L**p*(*E*(*x,t*))

*|E|*^{1}^{q} sup

*t>*2*kr**|E*(*x,t*)*|*^{−}^{1}^{q}*f**L**p*(*E*(*x,*2*kr*))≈*A*2*.*
Since*M*^{P}_{Ω,α}*f**L**q*(*E*)*A*1+*A*2*,* by Lemma4.1, we arrive at (4.3).

Let *p*=1. The inequality (4.4) directly follows from (4.1).

Similarly to Lemma4.3and Theorem3.1the following lemma is also proved.

LEMMA4.4. *Suppose that the function*Ω*∈L*_{s}(*S*_{ρ})*,* 1*<s*∞*, is A**t**-homoge-*
*neous of degree zero. Then for p>s*^{} *and any ballE* =*E*(*x,r*)*, the inequality*

*M*^{P}_{Ω}*f**L**p*(*E*(*x,r*))*r*^{γ}^{q} sup

*t>*2*kr**t*^{−}^{γ}^{p}*f**L**p*(*E*(*x,t*))

*holds for f∈L*^{loc}_{p} (R^{n})*.*

THEOREM4.1. *Suppose that*0*<*α *<*γ *and the function*Ω*∈L* γ

γ*−*α(*S*_{ρ}) *is A**t**-*
*homogeneous of degree zero. Let* 1*p<* _{α}^{γ}*,* ^{1}_{q}= ^{1}_{p}*−*^{α}_{γ}*, and* (ϕ1*,*ϕ2) *satisfy the*
*condition*

*r<t<*∞sup *t*^{α−}^{γ}^{p} ess inf

*t<*τ*<*∞ϕ1(*x,*τ)τ^{γ}^{p} *C*ϕ2(*x,r*)*,* (4.5)
*where C does not depend on x and r . Then the operator M*_{Ω,α}^{P} *is bounded from*
*M*_{p,ϕ}_{1}*,P*(R^{n}) *to* *M*_{q,ϕ}_{2}*,P*(R^{n})*for p>*1 *and from* *M*_{1,ϕ}_{1}*,P*(R^{n})*to WM*_{q,ϕ}_{2}*,P*(R^{n}) *for*
*p*=1*.*

*Proof.* By Theorem2.4and Lemma4.3we get
*M*_{Ω,α}^{P} *f**M*_{q,}_{ϕ}_{2,P} sup

*x∈*R^{n}*,r>*0ϕ2(*x,r*)^{−1}sup

*t>r**t*^{−}^{γ}^{q}*f**L**p*(*E*(*x,t*))

sup

*x∈*R^{n}*,r>*0ϕ1(*x,r*)^{−1}*r*^{−}^{γ}^{p}*f**L**p*(*E*(*x,r*))=*f**M**p*<