Kato class (K_{n}^{infty }(D))

Definition 2.1

(See [28])

A function ψ in (mathcal{B}(D)) is said to be in the Kato class (K_{n}^{infty }(D)) if

$$ lim_{rrightarrow 0} biggl( sup_{xin D} int _{Dcap B(x,r)} Gamma (x,y) biglvert psi (y) bigrvert ,dy biggr) =0, $$

(2.1)

and

$$ lim_{Mrightarrow infty } biggl( sup_{xin D} int _{D cap ( vert y vert geq M)}Gamma (x,y) biglvert psi (y) bigrvert ,dy biggr) =0, $$

(2.2)

where (Gamma (x,y)) is given by (1.13).

Example 2.2

Let (p>frac{n}{2}). Then, we have

$$ L^{p} ( D ) cap L^{1} ( D ) subset K_{n}^{ infty }(D). $$

Indeed, for (psi in L^{p} ( D ) ), by using the Hölder inequality, it is clear that (2.1) holds. Now, assume further that (psi in L^{1} ( D ) ), then

$$begin{aligned}& int _{Dcap ( vert y vert geq M)}Gamma (x,y) biglvert psi (y) bigrvert ,dy \& quad leq int _{Dcap B(x,r)}Gamma (x,y) biglvert psi (y) bigrvert ,dy +c_{n}r^{2-n} int _{Dcap ( vert y vert geq M)cap ( vert x-y vert geq r)} biglvert psi (y) bigrvert ,dy. end{aligned}$$

Hence, ψ satisfies (2.2).

The next Lemma is due to Mâagli and Zribi, see [20, Remark 2 and Proposition 1].

Lemma 2.3

  1. (i)

    Let ψ be a radial function in D, then

    $$ psi in K_{n}^{infty }(D)quad textit{if and only if}quad int _{0}^{infty }r biglvert psi (r) bigrvert ,dr< infty . $$

  2. (ii)

    Let (psi in mathcal{B}(D)) satisfying (2.1). Then, for each (M>0), we have

    $$ int _{Dcap ( vert y vert leq M)} biglvert psi (y) bigrvert ,dy < infty . $$

Remark 2.4

For all (x,y,zin mathbb{R}^{n}), we have

$$ frac{Gamma (x,y)Gamma (y,z)}{Gamma (x,z)}leq 2^{n-3}c_{n} bigl( Gamma (x,y)+ Gamma (y,z) bigr) , $$

(2.3)

where (c_{n}=frac{Gamma (frac{n}{2}-1)}{4pi ^{frac{n}{2}}}).

Proposition 2.5

Let (psi in K_{n}^{infty }(D)), (x_{0}in mathbb{R}^{n}) and (hin mathcal{S}^{+} ( D ) ). Then, we have

$$ lim_{rrightarrow 0} biggl( sup_{xin D} frac{1}{h(x)} int _{Dcap B(x_{0},r)}Gamma (x,y)h(y) biglvert psi (y) bigrvert ,dy biggr) =0, $$

(2.4)

and

$$ lim_{Mrightarrow infty } biggl( sup_{xin D} frac{1}{h(x)}int _{Dcap ( vert y vert geq M)}Gamma (x,y)h(y) biglvert psi (y) bigrvert ,dy biggr) =0. $$

(2.5)

Proof

Since (hin mathcal{S}^{+} ( D ) ), then by [23, Theorem 2.1, p. 164], there exists a sequence ((h_{k})_{k}subset mathcal{B}^{+}(D)) such that

$$ h(y)=sup _{k} int _{D}Gamma (y,z)h_{k}(z),dy , quad text{for }yin D. $$

Therefore, we need to prove (2.4) and (2.5) only for (h(y)=Gamma (y,z)) uniformly in (zin D).

Let (r>0). By using Remark 2.4, there exists a constant (c>0), such that for all (x,y,zin D),

$$ frac{1}{h(x)} int _{Dcap B(x_{0},r)}Gamma (x,y)h(y) biglvert psi (y) bigrvert ,dy leq 2csup_{xi in D} int _{Dcap B(x_{0},r)} Gamma (xi ,y) biglvert psi (y) bigrvert ,dy . $$

(2.6)

For (varepsilon >0), by Definition 2.1, there exists (s>0) and (M>0) such that

$$ int _{Dcap B(x_{0},r)}Gamma (xi ,y) biglvert psi (y) bigrvert ,dy leq varepsilon +frac{c_{n}}{s^{n-2}} int _{Dcap B(x_{0},r)cap ( vert y vert leq M)} biglvert psi (y) bigrvert ,dy . $$

Using this fact, (2.6) and Lemma 2.3(ii), we obtain (2.4) by letting (rrightarrow 0).

Finally, note that assertion (2.5) follows by using similar arguments as above. □

Proposition 2.6

Let (psi in K_{n}^{infty }(D)) and (varrho _{0}(x):= frac{1+ vert x vert ^{n-2}}{ vert x vert ^{n-2}}). Then, the function

$$ v(x):=frac{1}{varrho _{0}(x)} int _{D}Gamma (x,y) varrho _{0}(y)psi (y) ,dy $$

is continuous on (mathbb{R}^{n}) with (lim _{ vert x vert rightarrow infty }v(x)=0). That is, (v(x)in C_{0}(mathbb{R}^{n})).

Proof

Let (psi in K_{n}^{infty }(D)) and (x_{0}in mathbb{R}^{n}). Since (varrho _{0}in mathcal{S}^{+} ( D ) ), then for (varepsilon >0), by Proposition 2.5, there exists (M>r>0), such that the following holds:

(i) If (x_{0}neq 0), then for (xin B(x_{0},frac{r}{2})cap D), we have

$$ biglvert v(x)-v(x_{0}) bigrvert leq frac{varepsilon }{2}+ int _{D_{0}cap ( vert y vert leq M)} bigglvert frac{1}{varrho _{0}(x)}Gamma (x,y)- frac{1}{varrho _{0}(x_{0})}Gamma (x_{0},y) biggrvert varrho _{0}(y) biglvert psi (y) bigrvert ,dy , $$

where (D_{0}=Dcap B^{c}(0,r)cap B^{c}(x_{0},r)).

Since ((x,y)mapsto frac{1}{varrho _{0}(x)}Gamma (x,y)) is continuous on (( B(x_{0},frac{r}{2})cap D ) times (D_{0}cap ( vert y vert leq M))), we obtain by Lemma 2.3 (ii) and Lebesgue’s dominated convergence theorem,

$$ int _{D_{0}cap ( vert y vert leq M)} bigglvert frac{1}{varrho _{0}(x)}Gamma (x,y)- frac{1}{varrho _{0}(x_{0})}Gamma (x_{0},y) biggrvert varrho _{0}(y) biglvert psi (y) bigrvert ,dy rightarrow 0quad text{as }xrightarrow x_{0}. $$

Hence, there exists (delta >0) with (delta <frac{r}{2}) such that if (xin B(x_{0},delta )cap D),

$$ int _{D_{0}cap ( vert y vert leq M)} bigglvert frac{1}{varrho _{0}(x)}Gamma (x,y)- frac{1}{varrho _{0}(x_{0})}Gamma (x_{0},y) biggrvert varrho _{0}(y) biglvert psi (y) bigrvert ,dyleq frac{varepsilon }{2}. $$

Hence, for (xin B(x_{0},delta )cap D), we have

$$ biglvert v(x)-v(x_{0}) bigrvert leq varepsilon . $$

That is,

$$ lim_{xrightarrow x_{0}}v(x)=v(x_{0}). $$

(ii) If (x_{0}=0) and (xin B(0,frac{r}{2})cap D), then we have

$$ biglvert v(x) bigrvert leq frac{varepsilon }{2}+ int _{Dcap B^{c}(0,r)cap ( vert y vert leq M)} frac{1}{varrho _{0}(x)}Gamma (x,y)varrho _{0}(y) biglvert psi (y) bigrvert ,dy . $$

Now, since (lim _{ vert x vert rightarrow 0}frac{1}{varrho _{0}(x)}Gamma (x,y)varrho _{0}(y)=0), for all (yin Dcap B^{c}(0,r)cap ( vert y vert leq M)), we deduce by similar arguments as above that

$$ lim _{ vert x vert rightarrow 0} v(x)=0=v(x_{0}). $$

(iii) It remains to prove that (lim _{ vert x vert rightarrow infty }v(x)=0).

To this end, let (xin D) such that (vert x vert geq M+1). Using Proposition 2.5 and Lemma 2.3 (ii), we deduce that

$$begin{aligned} biglvert v(x) bigrvert leq &frac{varepsilon }{2}+ frac{1+M^{n-2}}{r^{n-2}} int _{Dcap B^{c}(0,r)cap ( vert y vert leq M)}Gamma (x,y) biglvert psi (y) bigrvert ,dy \ leq &frac{varepsilon }{2}+ frac{c}{( vert x vert -M)^{n-2}}, end{aligned}$$

where c is some positive constant.

This implies that (lim _{ vert x vert rightarrow infty }v(x)=0). □

Karamata regular variation theory

Let us recall some basic properties of Karamata regular variation theory (see [2, 14, 21, 24, 25]).

The following result concerns operations that preserve slow variation.

Proposition 2.7

If (L_{1}(t)), (L_{2}(t)) are slowly varying at infinity (resp., at zero), then the same holds for (L_{1}(t)+L_{2}(t)), (L_{1}(t)L_{2}(t)), and ((L_{1}(t))^{nu }) for any (nu in mathbb{R}).

Proposition 2.8

  1. (i)

    If (L(t)in mathcal{NSV}_{infty }), then for any (varepsilon >0)

    $$ lim _{trightarrow infty }t^{varepsilon }L(t)=infty ,qquad lim _{trightarrow infty }t^{-varepsilon }L(t)=0. $$

  2. (ii)

    If (L(t)in mathcal{NSV}_{0}), then for any (varepsilon >0 )

    $$ lim _{trightarrow 0^{+}}t^{varepsilon }L(t)=0 quad textit{and}quad lim_{trightarrow 0^{+}} t^{-varepsilon }L(t)=infty . $$

The following result, termed Karamata’s integration theorem, will be used later.

Proposition 2.9

Let (L(t)in mathcal{NSV}_{infty }). Then,

  1. (i)

    if (nu >-1),

    $$ int _{A}^{t}s^{nu }L(s),dssim frac{1}{nu +1}t^{nu +1}L(t),quad trightarrow infty ; $$

  2. (ii)

    if (nu <-1),

    $$ int _{t}^{infty }s^{nu }L(s),dssim – frac{1}{nu +1}t^{nu +1}L(t),quad trightarrow infty ; $$

  3. (iii)

    if (nu =-1),

    $$ l(t)= int _{A}^{t}s^{-1}L(s),dsin mathcal{NSV}_{infty }quad textit{and}quad lim _{trightarrow infty } frac{L(t)}{l(t)}=0; $$

and if (int _{A}^{infty }s^{-1}L(s),ds<infty ),

$$ m(t)= int _{t}^{infty }s^{-1}L(s),dsin mathcal{NSV}_{infty } quad textit{and}quad lim _{trightarrow infty } frac{L(t)}{m(t)}=0. $$

The following is an analog of Proposition 2.9 for L defined at zero instead of ∞.

Proposition 2.10

Let (L(t)in mathcal{NSV}_{0}). Then,

  1. (i)

    if (nu >-1),

    $$ int _{0}^{t}s^{nu }L(s),dssim frac{1}{nu +1}t^{nu +1}L(t),quad trightarrow 0^{+}; $$

  2. (ii)

    if (nu <-1),

    $$ int _{t}^{a}s^{nu }L(s),dssim – frac{1}{nu +1}t^{nu +1}L(t),quad trightarrow 0^{+}; $$

  3. (iii)

    if (nu =-1),

    $$ l_{0}(t)= int _{t}^{a}s^{-1}L(s),dsin mathcal{NSV}_{0}quad textit{and} quad lim _{trightarrow 0^{+}} frac{L(t)}{l_{0}(t)}=0; $$

and if (int _{0}^{a}s^{-1}L(s),ds<infty ),

$$ m_{0}(t)= int _{0}^{t}s^{-1}L(s),dsin mathcal{NSV}_{0}quad textit{and}quad lim_{trightarrow 0^{+}}frac{L(t)}{m_{0}(t)}=0. $$

The following result, will play a central role in establishing our main result in Sect. 3.

Proposition 2.11

For (alpha leq n) and (beta geq 2), set

$$ b(x)= vert x vert ^{-alpha }L_{0}bigl( vert x vert wedge 1bigr) bigl( vert x vert +1bigr)^{alpha -beta }L_{ infty } bigl( vert x vert vee 1bigr), quad xin D, $$

where (L_{0}in mathcal{NSV}_{0}) defined on ((0,a]), for some (a>1) and (L_{infty }in mathcal{NSV}_{infty }), defined on ([1,infty )) such that

$$ int _{0}^{a}s^{n-alpha -1}L_{0}(s),ds< infty quad textit{and}quad int _{1}^{ infty }s^{1-beta }L_{infty }(s),ds< infty . $$

(2.7)

Then,

$$ mathcal{N}b(x)approx vert x vert ^{min (0,2-alpha )}widetilde{L}_{0}bigl( vert x vert wedge 1bigr) bigl( vert x vert +1bigr)^{max (2-n,2-beta )-min (0,2-alpha )} widetilde{L}_{infty }bigl( vert x vert vee 1bigr),quad textit{on }D, $$

where for (tin (0,a)),

$$ widetilde{L}_{0}(t):=textstylebegin{cases} 1, & textit{if }alpha < 2, \ int _{t}^{a}frac{L_{0}(s)}{s},ds, & textit{if }alpha =2, \ L_{0}(t), & textit{if }2< alpha < n, \ int _{0}^{t}frac{L_{0} ( s ) }{s},ds, & textit{if }alpha =n,end{cases} $$

and for (tgeq 1),

$$ widetilde{L}_{infty }(t):=textstylebegin{cases} 1, & textit{if }beta >n, \ int _{1}^{t+1}frac{L_{infty }(s)}{s},ds, & textit{if }beta =n, \ L_{infty }(t+1), & textit{if }2< beta < n, \ int _{t+1}^{infty }frac{L_{infty } ( s ) }{s},ds, & textit{if }beta =2.end{cases} $$

Proof

Since b is a nonnegative radial measurable function on D, it follows from [23, Proposition 1.7], that

$$ mathcal{N}b(x):= int _{D}Gamma (x,y)b(y),dy =c int _{0}^{infty } frac{r^{n-1}}{ ( vert x vert vee r ) ^{n-2}}b(r),dr=:cJbigl( vert x vert bigr), $$

where the function J is defined on ([0,infty )) by

$$ J(t)= int _{0}^{infty } frac{r^{n-alpha -1}}{ ( tvee r ) ^{n-2}}(r+1)^{alpha-beta }L_{0}(r wedge 1)L_{infty }(rvee 1),dr. $$

We need to estimate (J(t)). Note that, under condition (2.7), (J(t)<infty ).

Let (a>1), then we have

$$begin{aligned} J(t) approx & int _{0}^{a} frac{r^{n-alpha -1}}{ ( tvee r ) ^{n-2}}L_{0}(r),dr+ int _{a}^{infty } frac{r^{n-beta -1}}{ ( tvee r ) ^{n-2}}L_{infty }(r),dr \ :=&J_{1}(t)+J_{2}(t). end{aligned}$$

We discuss the following cases:

Case 1. (0< tleq 1). Clearly from (2.7), we have

$$ J_{2}(t)= int _{a}^{infty }r^{1-beta }L_{infty }(r),dr approx 1. $$

On the other hand, by writing

$$ J_{1}(t)=t^{2-n} int _{0}^{t}r^{n-alpha -1}L_{0}(r),dr+ int _{t}^{a}r^{1- alpha }L_{0}(r),dr, $$

we deduce that

$$ J(t)approx t^{2-n} int _{0}^{t}r^{n-alpha -1}L_{0}(r),dr+ biggl(1+ int _{t}^{a}r^{1- alpha }L_{0}(r),dr biggr). $$

Therefore, by (2.7) and Propositions 2.8 and 2.10, we obtain

$$ J(t)approx phi _{0}(t):=textstylebegin{cases} 1, & text{if } alpha < 2, \ int _{t}^{a}frac{L_{0}(r)}{r},dr, & text{if } alpha =2, \ t^{2-alpha }L_{0}(t), & text{if } 2< alpha < n, \ t^{2-alpha }int _{0}^{t}frac{L_{0} ( r ) }{r},dr, & text{if } alpha =n. end{cases} $$

That is,

$$ J(t)approx t^{min (0,2-alpha )}widetilde{L}_{0}(t),quad text{for }0< t leq 1. $$

(2.8)

Case 2. (tgeq a+1). From (2.7), we derive that

$$ J_{1}(t)approx t^{2-n} int _{0}^{a}r^{n-alpha -1}L_{0}(r),dr approx t^{2-n}. $$

On the other hand, since

$$ J_{2}(t)=t^{2-n} int _{a}^{t}r^{n-beta -1}L_{infty }(r),dr+ int _{t}^{ infty }r^{1-beta }L_{infty }(r),dr, $$

we deduce that

$$ J(t)approx t^{2-n}biggl(1+ int _{a}^{t}r^{n-beta -1}L_{infty }(r),dr biggr)+ int _{t}^{infty }r^{1-beta }L_{infty }(r),dr. $$

Hence, by (2.7) and Propositions 2.8 and 2.9, we obtain

$$ J(t)approx textstylebegin{cases} t^{2-n}, & text{if }beta >n, \ t^{2-n}int _{a}^{t}frac{L_{infty }(s)}{s},ds, & text{if }beta =n, \ t^{2-beta }L_{infty }(t), & text{if }2< beta < n, \ int _{t}^{infty }frac{L_{infty } ( s ) }{s},ds, & text{if }beta =2.end{cases} $$

Therefore, by using Proposition 2.9 and [5, Lemma 2.3], we conclude that

$$ J(t)approx phi _{infty }(t):=textstylebegin{cases} (t+1)^{2-n}, & text{if }beta >n, \ (t+1)^{2-n}int _{1}^{t+1}frac{L_{infty }(s)}{s},ds, & text{if } beta =n, \ (t+1)^{2-beta }L_{infty }(t+1), & text{if }2< beta < n, \ int _{t+1}^{infty }frac{L_{infty } ( s ) }{s},ds, & text{if }beta =2.end{cases} $$

Hence,

$$ J(t)approx (t+1)^{max (2-n,2-beta )}widetilde{L}_{infty }(t),quad text{for }t geq a+1. $$

(2.9)

Finally, since (J(t)), (phi _{0}(t)) and (phi _{infty }(t)) are positive continuous functions on ([1,a+1]), we deduce that

$$ J(t)approx phi _{0}(t)phi _{infty }(t),quad text{on }[1,a+1]. $$

(2.10)

Hence, by combining (2.8), (2.9) and (2.10), we obtain

$$ J(t)approx t^{min (0,2-alpha )}widetilde{L}_{0}(twedge 1) (t+1)^{ max (2-n,2-beta )-min (0,2-alpha )}widetilde{L}_{infty }(tvee 1), quad text{on }[0,infty ). $$

This completes the proof. □

Proposition 2.12

Assume that p satisfies hypothesis (H), then

$$ mathcal{N}bigl(ptheta ^{gamma }bigr) (x)approx theta (x),quad textit{on }D, $$

where (gamma <1) and θ is given in (1.10).

Proof

Using (1.7) and (1.10), we obtain

$$begin{aligned}& p(x)theta ^{gamma }(x) \& quad thickapprox vert x vert ^{- alpha }mathcal{L}_{0}bigl( vert x vert wedge 1bigr) bigl( widetilde{mathcal{L}}_{0}bigl(min bigl( vert x vert ,1bigr)bigr) bigr) ^{ frac{gamma }{1-gamma }}bigl( vert x vert +1 bigr)^{alpha -beta }mathcal{L}_{infty }bigl( vert x vert vee 1 bigr) bigl( widetilde{mathcal{L}}_{ infty }bigl( vert x vert vee 1bigr) bigr) ^{ frac{gamma }{1-gamma }}, end{aligned}$$

where (alpha :=mu -gamma min (0,frac{2-mu }{1-gamma })) and (beta :=lambda -gamma max (2-n,frac{2-lambda }{1-gamma })).

From the fact that (mu leq n+(2-n)gamma ) and (lambda geq 2), we derive that (alpha leq n) and (beta geq 2).

By using the basic properties of Karamata regular variation theory and Proposition 2.11 with (L_{0}=mathcal{L}_{0}( vert x vert wedge 1) ( widetilde{mathcal{L}}_{0}(min ( vert x vert ,1)) ) ^{frac{gamma }{1-gamma }}in mathcal{NSV}_{0}) and (L_{infty }=mathcal{L}_{infty }( vert x vert vee 1) ( widetilde{mathcal{L}}_{infty }( vert x vert vee 1) ) ^{frac{gamma }{1-gamma }}in mathcal{NSV}_{infty }), we deduce that

$$ mathcal{N}bigl(ptheta ^{gamma }bigr) (x)approx vert x vert ^{ min (0,2-alpha )}widetilde{L}_{0}bigl( vert x vert wedge 1 bigr) bigl( vert x vert +1bigr)^{max (2-n,2-beta )-min (0,2- alpha )}widetilde{L}_{infty } bigl( vert x vert vee 1bigr). $$

Since (min (0,2-alpha )=min (0,frac{2-mu }{1-gamma }):=xi ) and (max (2-n,2-beta )=max (2-n,frac{2-lambda }{1-gamma }):=zeta ), we deduce that

$$ mathcal{N}bigl(ptheta ^{gamma }bigr) (x)thickapprox vert x vert ^{xi }widetilde{L}_{0}bigl( vert x vert wedge 1 bigr) bigl( vert x vert +1bigr)^{zeta -xi }widetilde{L}_{infty } bigl( vert x vert vee 1bigr)thickapprox theta (x). $$

This completes the proof. □

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