Growth properties for generalized Riesz potentials in central Herz-Morrey spaces (The structure of function spaces and its environment)

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(1)144. 数理解析研究所講究録第2041巻 2017年 144-153. Growth properties for generalized Riesz potentials in central Herz‐Morrey spaces Yoshihiro Mizuta. Abstract Riesz. decomposition theorem. says that. a. superharmonic function. on. the punc‐. tured unit ball B_{0} is represented as the sum of a generalized potential and a harmonic function outside the origin. Our first aim in this note is to study growth properties the. origin for generalized Riesz potentials of functions in central Herz‐Morrey B_{0}. We know another Riesz decomposition theorem which says that a superharmonic function on the unit ball B is represented as the sum of another generalized potential near. spaces. on. and. harmonic function. a. B. on. .. Our second aim in this note is to obtain. growth. properties near the boundary \partial B for generalized Riesz potentials of functions in central Herz‐Morrey spaces on B. A continuous function. Lebesgue [18]. if for every. u. on. an. \displaystyle \mathrm{m}_{\frac{\mathrm{a} {G} \mathrm{x}u=\max u\partial G Harmonic functions. $\Omega$. on. are. partial differential equations lems may be monotone. properties for. open set $\Omega$ is called monotone in the. relatively compact. open set. and. monotone in $\Omega$. sense. of. G\subset $\Omega$,. \displaystyle \mathrm{m}_{\frac{\mathrm{i} {G} \mathrm{n}u=\min_{\partial G}u. .. More. generally,. solutions of. elliptic prob‐ with growth. of second order and weak solutions for variational. (see [15]).. Our final aim in this note is concerned. monotone Sobolev functions in central. Herz‐Morrey. spaces.. Contents I. Isolated. Singularities. 1. Generalized Riesz. 2. Central. 3. Sobolevs. 4. Growth. II. 2. potentials. Herz‐Morrey. 2. 3. spaces. inequality. near. the. origin. 3. of. spherical. means. Boundary growth properties 1. 5. 6.

(2) 145. 5. Superharmonic. 6. Isolated. functions. singularities. B. on. 6. for monotone functions in the. sense. of. Lebesgue. [18, 1907]. 7. Part I. Isolated. Singularities. Generalized Riesz potentials. 1 Let. consider the Riesz kernel. us. I_{ $\alpha$}(x)=|x|^{ $\alpha$-n}. of order. $\alpha$. and. a. generalized kernel. I_{$\alpha$,\el}(x,y)=I_{$\alpha$}(x-y)-\displaystyle\sum_{|$\lambda$|\leql}\frac{(-y)^{$\lambda$}{$\lambda$!}D^{$\lambda$}I_{$\alpha$}(x) for. integer \ell ; when \ell\leq -1, I_{ $\alpha$,\ell}(x, y)=I_{ $\alpha$}(x-y). an. remainder of. |x+z|^{ $\alpha$-n}. We define the on. .. Note here that. around z=0.. generalized Riesz potential B_{0} by. of order. $\alpha$. for. a. I_{ $\alpha$,\ell}(x, y). is. Taylors. locally integrable function f. the puncture unit ball. I_{ $\alpha$,\el }f(x)=\displaystyle \int_{B_{0} I_{ $\alpha$,\el }(x, y)f(y)dy. Here. prepare the estimates for. we. LEMMA 1.1. (cf. [12,. Lemma. 3.2]).. (1) |I_{ $\alpha$,\ell}(x, y)| \leq C|x-y|^{ $\alpha$-n}. (2) |I_{ $\alpha$,\ell}(x, y)| \leq C|y|^{\ell+1}|x|^{ $\alpha$-n-l-1}. REMARK 1.2. If a. potential Let. u. and. be. r^{a}S(|u|, r). a. a. u. is. a. u. is. when. is bounded in. represented. (0,1). for. near. the. |x|/2.. <. 2|x| <|y|.. (see. some. on. B , then. u. is. represented. as. the. [2], [3], [14], [23]).. e.g.. on. B_{0} In view of Theorems .. 1.3 and 3.4 in. sum. [12],. of if. a>n-2 , then. a+2-n $\mu$(A(0, r))<\infty (A(0, r)=B(0,2r)\backslash B(0, r)). as. u(x)=I_{2,\ell} $\mu$(x)+ origin (except. is the Riesz measure;. |y|. superharmonic function. superharmonic function. \displaystyle \sup_{0<r<1/2}r and. |x|/2<|y| <2|x|.. when. harmonic function. kernels.. \ell\geq 0.. Let. when. (3) |I_{ $\alpha$,\ell}(x, y)| \leq C|y|^{\ell}|x|^{ $\alpha$-n-\ell}. generalized Riesz. at the. see. also. a. harmonic function. origin), where (2-n+a)-1<l\leq 2-n+a [8], [10], [11]. 2. and. $\mu$=c(- $\Delta$)u.

(3) 146. Central. 2. Herz‐Morrey. For 1 \leq p<\infty and a real number f on B satisfying. spaces consider the. v , we. family M^{p,q, $\nu$}(B). of all measurable. functions. \displaystyle \Vert f\Vert_{M(B)}p,q, $\nu$= (\int_{0}^{1}(r^{ $\nu$}\Vert f\Vert_{L^{p}(A(0,r) })^{q}dr/r)^{1/q}<\infty. when 0<q<\infty and. \displaystyle \Vert f\Vert_{M^{p,\infty, $\nu$}(B)}=\sup_{0<r<1} r^{ $\nu$}\Vert f\Vert_{L^{p}(A(0,r) } <\infty f=0 outside B as before; 0<q_{1} <q_{2}<\infty then. set. If. see. e.g.. ;. [4], [5], [16].. ,. M^{p,q_{1}, $\nu$}(B)\subset M^{p,q_{2}, $\nu$}(B)\subset M^{p,\infty, $\nu$}(B) Our space is somewhat. family of functions with finite weighted. a. mixed. norm. \displaystyle \Vert f\Vert_{p,q, $\omega$}= (\int(\int|f(x, y)|^{p} $\omega$(x, y)dy)^{q/p}dx)^{1/q}<\infty. Sobolevs. 3. p^{\mathrm{t}. Let. inequality. be the Sobolev exponent of p>1. :. 1/p^{\#}=1/p- $\alpha$/n>0. LEMMA 3.1. (Sobolevs inequality (cf. [1], [23])).. There is. a. constant C>0 such that. \Vert I_{ $\alpha$}f\Vert_{Lp}t_{(\mathbb{R}^{n})} \leq C\Vert f\Vert_{L^{\mathrm{p} (\mathbb{R}^{n})}. Sobolevs. Herz‐Morrey. inequality. is extended to. generalize. Riesz. potentials of functions. in central. spaces.. $\alpha$-n/p< $\nu$<n-n/p. THEOREM 3.2. Assume that. .. Then. \Vert I_{ $\alpha$}f\Vert_{M^{\mathrm{p}\mathrm{q}, $\nu$}(B)}\#, \leq C\Vert f\Vert_{M^{p,q, $\nu$}(B)}. (Sobolevs inequality for generalized n-n/p+P< $\nu$<n-n/p+\ell+1 Then. THEOREM 3.3. and. Riesz. potentials).. Assume that \ell\geq 0. .. \Vert I_{ $\alpha$,\ell}f\Vert_{M^{pq, $\nu$}(B)}\#,\leq C\Vert f\Vert_{M(B)}p,q, $\nu$. To prove Sobolevs. inequality,. for. a. real number. $\beta$ and. 0<r. Hardy type operators. H_{ $\beta$}^{-}f(r)=r^{- $\beta$}\displaystyle \int_{B(0,r)}|y|^{ $\beta$-n}f(y)dy. and. for measurable functions. f. H_{ $\beta$}^{+}f(r)=r^{- $\beta$}\displaystyle \int_{B\backslash B(0,r)}|y|^{ $\beta$-n}f(y)dy on. B.. 3. < 1,. let. us. consider the.

(4) 147. $\beta$- $\nu$-n/p> $\epsilon$>0. LEMMA 3.4. Let. Then. .. H_{ $\beta$}^{-}f(r)\displaystyle \leq Cr^{- $\epsilon$-n/p- $\nu$}(\int_{0}^{r}(t^{ $\epsilon$+\mathrm{v} \Vert f\Vert_{L\mathrm{p}(A(0,t) })^{q}\frac{dt}{t})^{1/q} for all 0<r<1 and LEMMA 3.5. Let. f\in L_{1\mathrm{o}\mathrm{c} ^{1}(\mathbb{R}^{n}). .. 0< $\epsilon$<- $\beta$+ $\nu$+n/p. .. Then. H_{ $\beta$}^{+}f(r)\displaystyle \leq Cr^{ $\epsilon$-n/p- $\nu$}(\int_{r/2}^{1}(t^{- $\epsilon$+\mathrm{v} \Vert f\Vert_{Lp(A(0,t) })^{q}\frac{dt}{t})^{1/q} for all 0<r< 1 and. Proof of. f\in L_{1\mathrm{o}\mathrm{c} ^{1}(\mathbb{R}^{n}). Theorem 3.3. Let. .. \Vert f\Vert_{M(B)}\mathrm{p},q, $\nu$. \leq 1 and f\geq 0 For x\in B , set .. I_{ $\alpha$,\el }f(x) = \displaystyle \int_{B(0,|x|/2)}I_{ $\alpha$,\el }(x, y)f(y)dy +\displaystyle \int_{B(0,2|x|)\backslash B(0,|x|/2)}I_{ $\alpha$,l}(x, y)f(y)dy +\displaystyle \int_{B(0,1)\backslash B(0,2|x|)}I_{ $\alpha$,\el }(x, y)f(y)dy = u_{1}(x)+u_{2}(x)+u_{3}(x). Let 0<r<1. By. .. Lemma 1.1. we. .. have. |u_{2}(x)| \displaystyle \leq C\int_{A(0,r/2)\cup A(0,r)}|x-y|^{ $\alpha$-n}|f(y)|dy for. x\in A(0, r). ,. so. that Lemma 3.4. gives. \Vert u_{2}\Vert_{L^{p}(A(0,r))}\# \leq C\Vert f\Vert_{L^{p}(A(0,r/2)\cup A(0,r))}\#. Hence,. \displaystyle \int_{0}^{1}(r^{$\nu$}\Vert u_{2}\Vert_{L^{p(\cdot)}(A(0,r) })^{q}\frac{dr}{r} By. Lemma 1.1. we see. \leq. C\displaystyle \int_{0}^{1}(t^{ $\nu$}\Vert f\Vert_{Lp(A(0,r/2)\cup A(0,r) })^{q}\frac{dt}{t}.. that. |u_{1}(x)| \displaystyle \leq C|x|^{ $\alpha$-n-\ell-1}\int_{B(0,|x|/2)}|y|^{l+1}f(y)dy \leq Cr^{ $\alpha$}H_{n+l+1}^{-}f(r). for. x\in A(0, r) Hence, using .. Lemma. 3.4,. we. find. \displaystyle \Vert u_{1}\Vert_{L^{p(\cdot)}(A(0,r) } \leq Cr^{- $\epsilon$- $\nu$}(\int_{0}^{r}(t^{ $\epsilon$+ $\nu$}\Vert f\Vert_{L^{p}(A(0,t) })^{q}\frac{dt}{t})^{1/q} 4.

(5) 148. for. 0< $\epsilon$<n+\ell+1-n/p- $\nu$ Consequently, .. \displaystyle \int_{0}^{1}(r^{ $\nu$}\Vert u_{1}\Vert_{L^{p}(A(0,r) })^{q}\frac{dr}{r} \leq C\int_{0}^{1}(r^{- $\epsilon$}\int_{0}^{r}(t^{ $\epsilon$+\mathrm{v} \Vert f\Vert_{L^{p}(A(0,t) })^{q}\frac{dt}{t})\frac{dr}{r} \displaystyle \leq C\int_{0}^{1}(t^{ $\epsilon$+ $\nu$}\Vert f\Vert_{L^{p}(A(0,t) })^{q}(\int_{t}^{1}r^{- $\epsilon$ q}\frac{dr}{r})\frac{dt}{t} \displaystyle \leq C\int_{0}^{1}(t^{\mathrm{v} \Vert f|_{L^{p}(A(0,t) })^{q}\frac{dt}{t}. Similarly, by. Lemma 1.1. we see. that. |u_{3}(x)| \displaystyle \leq C|x|^{ $\alpha$-n-\el }\int_{B\backslash B(0,2|x|)}|y|^{\el }f(y)dy \leq Cr^{ $\alpha$}H_{n+l}^{+}f(r). for. for. x\in A(0, r) Hence, using .. Lemma 3.5,. we. find. \displaystyle\Vertu_{3}\Vert_{L^{p}(A(0,r) }\leqCr^{$\epsilon$-$\nu$}(\int_{r}^{1}(t^{-$\epsilon$+$\nu$}\Vertf\Vert_{L\mathrm{p}(A(0,t) })^{\mathrm{q} \frac{dt}{t})^{1/q}. 0< $\epsilon$<-(n+\ell-n/p-\mathrm{v}) Thus, .. \displaystyle \int_{0}^{1}(r^{\mathrm{v} \Vert u_{3}\Vert_{L^{\mathrm{p}(\cdot)}(A(0,r) })^{q}\frac{dr}{r} \leq C\int_{0}^{1} (r^{ $\epsilon$}\int_{r}^{1}(t^{- $\epsilon$+\mathrm{v} \Vert f\Vert_{L^{p}(A(0,t) })^{q}\frac{dt}{t}) \frac{dr}{r} \displaystyle \leq C\int_{0}^{1}(t^{- $\epsilon$+\mathrm{v} \Vert f\Vert_{Lp(A(0,t) })^{q}(\int_{0}^{t}r^{ $\epsilon$ q}\frac{dr}{r}) \frac{dt}{t} \displaystyle \leq C\int_{0}^{1}(t^{ $\nu$}\Vert f\Vert_{L^{\mathrm{p} (A(0,t) })^{q}\frac{dt}{t} Growth. 4. The L^{q}. near. (1 \leq q <\infty). the. means over. of. origin the. spherical. \square. means. spherical surface S(0, r) for. a. function. u. is defined. by. where. S_{q}(u, r) = (\displaystyle \frac{1}{|S(0,r)|}\int_{S(0,r)} |u(x)|^{q}dS(x) ^{1/q} = (\displaystyle \frac{1}{$\omega$_{n-1} \int_{S(0,1)} |u(r $\sigma$)|^{q}dS( $\sigma$) ^{1/q}. S(0, r)=\partial B(0, r). sphere.. and. |S(0, r)|=$\omega$_{n-1}r^{n-1}. with $\omega$_{n-1}. denoting. Our aim is to find d>0 such that. \displaystyle \lim_{r\rightar ow 0}\inf_{+} r^{d}S_{q}(I_{ $\alpha$,\ell}f, r)=0 for. a. function. f. Our result is. on a. B. satisfying Herz‐Morrey type. continuation of Gardiners result 5. conditions.. ([13, 1988]). :. the. area. of the unit.

(6) 149. REMARK 4.1. For. (1). if. a. Green potential. G $\mu$. B,. on. (n-1)/(n-2) \leq q<(n-1)/(n-3). ,. then. \displaystyle \lim_{r\rightarrow}\inf_{1} (1-r)^{n-1-(n-1)/q}S_{q}(G $\mu$, r)=0 ; (2). if. 1\leq q<(n-1)/(n-2). ,. then. \displaystyle \lim_{r\rightarrow 1} (1-r)^{n-1-(n-1)/q}S_{q}(G $\mu$, r)=0. THEOREM 4.2.. 1/q\leq 1/p. <. THEOREM 4.3.. 1/q\leq 1/p. Suppose n-n/p+\ell< $\nu$<n-n/p+\ell+1 If (n- $\alpha$ p-1)/(p(n-1)). <. .. If. \displaystyle \lim_{r\rightar ow 0}\inf_{+} r^{(n- $\alpha$ p+ $\nu$ p)/p}S_{q}(I_{ $\alpha$,\ell}f, r)<\infty. f\in M^{p, $\nu$}(B). for all. for all. (n- $\alpha$ p-1)/(p(n-1)). Suppose n-n/p+l< $\nu$<n-n/p+\ell+1. then. ,. .. .. then. ,. \displaystyle \lim_{r\rightar ow 0}\inf_{+} r^{(n- $\alpha$ p+ $\nu$ p)/p}S_{q}(I_{ $\alpha$,\ell}f, r)=0. f\in M_{0}^{p, $\nu$}(B). .. Part II. Boundary growth properties Superharmonic functions. 5. The Riesz kernel is written. on. B. as. |x-y|^{ $\alpha$-n}=\displaystyle \sum_{l}(1-|y|)^{\el }$\phi$_{ $\alpha$,\el } ( where ỹ. =. y/ | y |. x,. ỹ),. and. $\phi$_{$\alpha$,\el}(x,\displaystyle\ovalbox{\t \smal REJECT})=\sum_{\el/2\leqk\leq\el}a_{$\alpha$,\el,k}|x-\tilde{y}|^{$\alpha$-n 2k}(x.\ovalbox{\t \smal REJECT}-1)^{2k-l}. In. fact, consider. the. Taylor expansion of. |x-y|^{ $\alpha$-n}=|x-\ovalbox{\tt\small REJECT}+ t\tilde{y}|^{ $\alpha$-n} and set. t=1-|y|.. Now define. K_{$\alpha$,m}(x,y)=\displayst le\frac{1}(n-$\alpha$) \sigma$_{n}\left\{ begin{ar y}{l |x-y^{$\alpha$-n}&(y\inB(0,1/2) ;\ |x-y^{$\alpha$-n}\sum_{\el=0}^{m}(1-|y)^{\el}$\phi$_{$\alpha$,l}(x,\ovalbox{\t smal REJ CT})&(y\inB\backslahB(0,1/2) . \end{ar y}\right. The. following properties for K_{2,m}. are. fundamental. 6.

(7) 150. LEMMA 5.1. (cf. [9,. Lemma. 2.2]).. (1) $\Delta$ K_{2,m}(\cdot, y)=$\delta$_{y}. (2) |K_{ $\alpha$,m}(x, y)| \leq C|x-y|^{ $\alpha$-n-m-1}(1-|y|)^{m+1} We show Riesz. when. when n>2 ;. 1-|y|\leq ( \sqrt{}2—1) | x—ỹ |. decomposition for superharmonic functions. THEOREM 5.2. If u is. superharmonic. on. .. B.. in B and. \displaystyle \lim_{r\rightarrow}\inf_{1}(1-r)^{a}S(u, r)>-\infty, then. u(x)=\displaystyle \int_{B}K_{2,m}(x, y) d $\mu$(y)+h_{0}(x) where h_{0} is harmonic in B and. m. is. an. integer greater than. ,. a.. Set. C(0, r)=B(0, r+(1-r)/2)\backslash B(0, r-(1-r)/2) for 0<r<1. Denote. by. \tilde{M}^{p, $\nu$}(B). the. family of. all functions. f\in L_{1\mathrm{o}\mathrm{c} ^{1}(B). such that. \displaystyle \Vert f\Vert_{M^{p, $\nu$}(B)}-=\sup_{0<r<1}(1-r)^{ $\nu$}\Vert f\Vert_{Lp(C(0,r) } <\infty. Now. we. give. a. continuation of the results. by Gardiner [13].. THEOREM 5.3. Let 1 \leq q < \infty and suppose Then there exists a constant C>0 such that. (1). if n+m-1-. \Vert F\Vert_{M^{\mathrm{p},u}(B)}-. $\alpha$ p<(n-1)/q\leq n+m- $\alpha$ p. \leq 1 with. F(y). =. (1-|y|)f(y). .. then. ,. \displaystyle \lim_{r\rightar ow}\inf_{1}(1-r)^{(n- $\alpha$ p+ $\nu$)/p-(n-1)/q}S_{q}(K_{ $\alpha$,m}f, r)\leq C ; (2). if n-. $\alpha$ p+m<(n-1)/q<n+m+1- $\alpha$ p. then. ,. 1/2<r<1\displaystyle \sup(1-r)^{(n- $\alpha$ p+ $\nu$)/p-(n-1)/q}S_{q}(K_{ $\alpha$,m}f, r)\leq C. 6. Isolated sense. of. singularities for monotone Lebesgue [18, 1907]. A continuous function. [18]. u on a. if for every subdomain. domain D is said to be monotone in the. Heinonen‐Kilpeläinen‐. [20, 21], [30, 31]. the author. Martio. [22, 23],. sense. of. G, \overline{G}\subset D,. \displaystyle \max_{\overline{G} u=\max u\partial G see. functions in the. [15],. and. \displaystyle \mathrm{m}_{\frac{\mathrm{i} {G} \mathrm{n}u=\min_{\partial G}u ;. Koskela‐Manfredi‐Villamor. the author‐Shimomura. 7. Lebesgue. [24, 25],. [17],. Manfredi‐Villamor. Villamor‐Li. [29]. ,. Vuorinen.

(8) 151. Suppose. THEOREM 6.1. monotone in the. sense. n-1 <p < $\nu$+n Let of Lebesgue and satisfies .. be. u. a. function. \displaystyle \sup_{0<r<1}r^{ $\nu$}\int_{A(0,r)}|\nabla u(x)|^{p}dx\leq 1 (p>n-1). on. B\backslash \{0\}. which is. .. Then. \displaystyle \sup_{x\in B} r^{(n-p- $\nu$)/p}|u(x)| \leq C<\infty. For monotone functions in LEMMA 6.2. sense. of Lebesgue, the. (cf. [20], [21], [23]). If u then \forall x, y\in B(x_{0}, r). and p_{1}>n-1. following. is monotone in. is. B(x_{0},2r). in the. sense. of. Lebesgue. ,. |u(x)-u(y)|^{p_{1} \displaystyle \leq Cr^{p_{1}-n}\int_{B(x0,2r)}|\nabla u(z)|^{p_{1} dz EXAMPLE 6.3. For. If. crucial tool.. a. $\beta$>0 consider ,. is monotone in. B\backslash \{0\}. \bullet. u. \bullet. |\nabla u(x)| \leq C|x|^{- $\beta$-1}. u(x)=|x|^{- $\beta$}. in the. sense. of. .. (6.1). .. Then. Lebesgue. ;. -( $\beta$+1)p+ $\nu$+n\geq 0,. \displaystyle \sup_{0<r<1}r^{ $\nu$}\int_{A(0,r)}|\nabla u(x)|^{p}dx<\infty. Hence, letting $\beta$=(n-p+ $\nu$)/p\geq 0. ,. we. find. \displaystyle \lim_{x\rightar ow 0} |x|^{(n-p+ $\nu$)/p}u(x)=1. Finally we show boundary growth for monotone functions THEOREM 6.4.. Suppose n-1<p< $\nu$+n, p<q<\infty sense of Lebesgue and satisfies. .. on. Let. u. B in the. be. a. is monotone in the. \displaystyle \sup_{0<r<1}(1-r)^{ $\nu$}\int_{C(0,r)}|\nabla u(x)|^{p}dx\leq 1. If. (n-1)/q<(n-p+\mathrm{v})/p. ,. then. \displaystyle \sup_{0<r<1}(1-r)^{(n-p- $\nu$)/p-(n-1)/q}S_{q}(u, r)\leq C<\infty.. 8. sense. function. of Lebesgue. on. B which.

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