Pointwise multipliers on Musielak-Orlicz and Musielak-Orlicz-Morrey spaces (Researches on isometries from various viewpoints)
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(2) 81. This. equality. extend the above. Morrey. equality. In this report. we. to Musielak‐Orlicz spaces and Musielak‐Orlicz‐. spaces.. Recall. that, for. E has the lattice. a. normed. that, if. It is known. quasi‐normed. or. (ideal) property. f\in E, h\in L^{0}( $\Omega$). if the. |h(x)|\leq|f(x)|. ,. \Vert g\Vert_{\mathrm{o}\mathrm{p}. is the. sider pointwise. operator. show the. norm. multipliers from. Musielak‐Orlicz‐Morrey For the. \Rightarrow. a. and. of g. ,. we. say that. holds:. h\in E, \Vert h\Vert_{E}\leq \Vert f\Vert_{E}.. complete,. then. \Vert g\Vert_{\mathrm{o}\mathrm{p} =\Vert g\Vert_{L\infty( $\Omega$)},. \in \mathrm{P}\mathrm{W}\mathrm{M}(E). In this report. .. Musielak‐Orlicz‐Morrey. we con‐. space to another. space.. introduction, first. following. following. a.e.. E\subset L^{0}( $\Omega$). space. E has the lattice property and is. \mathrm{P}\mathrm{W}\mathrm{M}(E)=L^{\infty}( $\Omega$) where. by [7, 8].. extended to Orlicz spaces. was. we. show the. proof of (1.1).. To do this. we. first. lemma.. Lemma 1.1.. g\in L^{p_{3} ( $\Omega$) \Rightarrow \Vert g\Vert_{\mathrm{o}\mathrm{p} =\Vert g\Vert_{L^{p_{3} ( $\Omega$)} g\in L^{\mathrm{P}3}( $\Omega$) Then, by L^{p_{1} ( $\Omega$) to L^{p_{2} ( $\Omega$) and. Proof. Let ator from. .. Hölders. inequality,. (1.2). .. g is. a. bounded oper‐. \Vert g\Vert_{\mathrm{o}\mathrm{p} \leq \Vert g\Vert_{L^{p_{3} ( $\Omega$)}. Let. f. =. |g|^{p_{3}/p_{1}}. fg\in If^{2}( $\Omega$). ,. Then. .. f. \in. IP^{1}( $\Omega$). \Vert fg\Vert_{L^{\mathrm{p}_{2} ( $\Omega$)}=\Vert g\Vert_{L^{p_{3} ( $\Omega$)}^{p_{3}/p_{2}. ,. and. \Vert f\Vert_{L^{p_{1} ( $\Omega$)}. =. and. \Vertg\Vert_{L^{\mathrm{p}_{3}($\Omega$)}^{p3/p_{1}. .. Moreover,. \Vert f\Vert_{L^{p_{1} ( $\Omega$)}\Vert g\Vert_{L^{p_{3} ( $\Omega$)}=\Vert fg\Vert_{L^{p_{2} ( $\Omega$)}, since. This shows that. To prove. (1.2).. (1.1). we. \displaystyle \frac{p_{3} {p_{1} +1=p_{3}(\frac{1}{p_{1} +\frac{1}{p_{3} ) =\frac{p_{3} {p_{2} .. \square. need to show. PWM ( Ư1. ( $\Omega$), L^{p_{2} ( $\Omega$))\subseteq L^{p_{3} ( $\Omega$). .. (1.3).
(3) 82. Proof of. (1.3).. Let. g\in \mathrm{P}\mathrm{W}\mathrm{M} ( Ư1 ( $\Omega$). simple functions g_{j}\geq 0. ,. Ư2 ( $\Omega$ )). Take. g_{j}\nearrow|g|. such that. a.e.. a. Then, for. sequence of any. finitely. f\in L^{p_{1}}( $\Omega$). ,. we. have. \Vert fg_{j}\Vert_{L^{p_{2} ( $\Omega$)}\leq\Vert fg\Vert_{L^{p_{2} ( $\Omega$)}. the uniform boundedness theorem and Lemma 1.1. By. and. \displaystyle \sup_{j}\Vert g_{j}\Vert_{\mathrm{o}\mathrm{p} <\infty Therefore, g\in IP^{3}( $\Omega$) Another. operator from. \displaystyle \sup_{j}\Vert g_{j}\Vert_{L^{p_{3} ( $\Omega$)}<\infty. \square. .. (1.3). Let g\in \mathrm{P}\mathrm{W}\mathrm{M}(L^{p_{1} ( $\Omega$), Ij^{p_{2} ( $\Omega$)) L^{p_{1} ( $\Omega$) to U^{2}( $\Omega$) Actually, if of. proof. have. we. Then g is. .. a. closed. .. $\Gamma$/. f_{j}\rightarrow f then. we can. take its. in. L^{p_{1} ( $\Omega$). By of. h=fg. the closed. finitely simple. PWM(Ư ( $\Omega$ ). ,. f_{j}g\rightarrow h. in. L^{p_{2} ( $\Omega$). ,. subsequence f_{j(k)} such that. f_{j(k)}\rightarrow f This shows that. and. a.e.. and. a.e.. f_{j(k)}g\rightarrow h. That is, g is. graph theorem. g is. functions g_{j} \geq. a. a. closed operator.. bounded operator. Take. 0 such that g_{j}. Ư2 ( $\Omega$ )) \cap\ovalbox{\t \smal REJECT} 3 ( $\Omega$ ) and. a.e.. then, by. \nearrow |g|. Lemma 1.1. a.e.. we. a. sequence. Then g_{j} \in. have. I L^{p_{3} ( $\Omega$)=\Vert g_{j}\Vert_{0_{\mathrm{P} \leq}\Vert g\Vert_{\mathrm{o}\mathrm{p} , for all j. 2. .. Therefore, g\in IP^{3}( $\Omega$). \square. .. Orlicz and Musielak‐Orlicz spaces. Let \overline{$\Phi$} be the set of all functions $\Phi$. \displaystyle \lim_{t\rightar ow+0} $\Phi$(t)= $\Phi$(0)=0. :. [0, \infty]\rightarrow[0, \infty]. and. such that. \displaystyle \lim_{t\rightar ow\infty} $\Phi$(t)= $\Phi$(\infty)=\infty.. Let. a( $\Phi$)=\displaystyle \sup\{t\geq 0: $\Phi$(t)=0\}, b( $\Phi$)=\inf\{t\geq 0: $\Phi$(t)=\infty\}..
(4) 83. Definition 2.1. A function $\Phi$\in \overline{$\Phi$} is called also called. [0, b( $\Phi$)). ,. an. Orlicz. function). if $\Phi$ is. a. Young function (or. pondecreasing. on. Any Young. function is neither. We denote. .. and. convex on. and. \displaystyle \lim_{t\rightar ow b( $\Phi$)-0} $\Phi$(t)= $\Phi$(b( $\Phi$) (\leq\infty) (0, \infty). [0, \infty ). sometimes. by $\Phi$_{\mathrm{Y}. identically. the set of all. We define three subsets. zero. .. identically infinity. nor. on. Young functions.. y^{(i)} (i=1,2,3). of Young functions. as. \mathcal{Y}^{(1)}=\{ $\Phi$\in $\Phi$_{Y}:b( $\Phi$)=\infty\}, \mathcal{Y}^{(2)}=\{ $\Phi$\in $\Phi$_{Y}:b( $\Phi$)<\infty, $\Phi$(b( $\Phi$))=\infty\}, \mathcal{Y}^{(3)}=\{ $\Phi$\in $\Phi$_{Y}:b( $\Phi$)<\infty, $\Phi$(b( $\Phi$))<\infty\}. See. Figure. 1.. Definition 2.2. (Orlicz space).. For. a. function $\Phi$\in $\Phi$_{\mathrm{y} , let. }, { f\in L^{0}( $\Omega$) \displaystyle \int_{ $\Omega$} $\Phi$(k|f(x)| d $\mu$(x)<\infty \displaystyle \Vert f\Vert_{L^{\dot{ $\Phi$} ( $\Omega$)}=\inf\{ $\lambda$>0:\int_{ $\Omega$} $\Phi$(\frac{|f(x)|}{ $\lambda$})d $\mu$(x)\leq 1\}.. L^{ $\Phi$}( $\Omega$)=. For. for. :. some. k>0. example. $\Phi$(t)=t^{p}(\in \mathcal{Y}^{(1)}) \Rightarrow L^{ $\Phi$}( $\Omega$)=L^{p}( $\Omega$). $\Phi$(t)=\left\{ begin{ar ay}{l} 0&(0\leqt\leq1)(\iny^{(3)} \ \infty&(t>1) \end{ar ay}\right.. \Rightarrow. ,. L^{ $\Phi$}( $\Omega$)=L^{\infty}( $\Omega$). .. To show. \mathrm{P}\mathrm{W}\mathrm{M}(L^{$\Phi$_{1} ( $\Omega$), L^{$\Phi$_{2} ( $\Omega$))=L^{$\Phi$_{3} ( $\Omega$) we. need. generalized. Hölders. ,. inequality. \Vert fg\Vert_{L^{$\Phi$_{2} ( $\Omega$)}\leq C\Vert f\Vert_{L^{$\Phi$_{1} ( $\Omega$)}\Vert g\Vert_{L^{$\Phi$_{3} ( $\Omega$)} and. \Vert g\Vert_{\mathrm{o}\mathrm{p} \sim\Vert g\Vert_{L^{$\Phi$_{3} ( $\Omega$)}. for. g\in L^{$\Phi$_{3} ( $\Omega$). .. (2.1).
(5) 84. t. Figure. 1: Three. types of Young functions.
(6) 85. If. we. prove. \displayst le\int_{$\Omega$} \Phi$_{3}(\frac{|g(x)|}{\Vertg\Vert_{L^$\Phi$_{3}($\Omega$)} d $\mu$(x)=1 then. get. we. (2.1). However,. restriction. So we. we. this holds if and. prove it for all. Definition 2.3. Let is. a. Young. $\Phi$_{Y}^{v}. measure, there exists. g\not\equiv 0,. only if $\Phi$_{3}\in$\Delta$_{2} which. is. ,. .. strong. To do this. Assume also. .. $\Phi$_{Y} for. in. Let. $\Phi$_{GY}^{v}. is in. t\in(0, \infty). (i) Let $\Phi$_{G\mathrm{Y} some P\in(0,1 ].. $\Phi$_{\mathrm{Y} ^{v}. $\Omega$\times. any subset A\subset $\Omega$ with finite t ) $\chi$_{A} is. be the set of all $\Phi$\in. :. integrable.. \overline{$\Phi$}. $\Omega$\times\cdot[0, \infty]\rightarrow[0, \infty]. such that. $\Phi$((\cdot)^{1/l}). such that $\Phi$. is. (\cdot)^{1/l} ). \ell\in(0,1 ].. some. example, let. :. that, for. such that $\Phi$. be the set of all $\Phi$ for. [0, \infty]\rightarrow[0, \infty] such that and that $\Phi$ t) is measurable on. be the set of all $\Phi$. Definition 2.4.. For. with. finitely simple functions g\not\equiv 0. function for every x\in $\Omega$ ,. [0, \infty]. $\Omega$ for every t\in. (ii). g\in L^{$\Phi$_{3} ( $\Omega$). $\Phi$_{3}\in \mathcal{Y}^{(1)}\cup y^{(2)}.. need. $\Phi$(x, \cdot). for all. $\Phi$(x, t)=t^{p(x)}. p_{-}\geq 1 \Rightarrow $\Phi$\in $\Phi$_{\mathrm{Y}}^{v},. p_{-}>0 \Rightarrow $\Phi$\in $\Phi$_{GY}^{v}. For. $\Phi$, $\Psi$\in \overline{$\Phi$}. ,. we. write $\Phi$\approx $\Psi$ if there exists. a. positive. constant C such. that. $\Phi$(C^{-1}t)\leq $\Psi$(t)\leq $\Phi$(Ct) For. $\Phi$, $\Psi$. :. $\Omega$\times[0, \infty]\rightarrow[0, \infty]. ,. we. for all. t\in(0, \infty). .. also write $\Phi$\approx $\Psi$ if there exists. a. positive. constant C such that. $\Phi$(x, C^{-1}t)\leq $\Psi$(x, t)\leq $\Phi$ ( x Ct) ,. Lemma 2.1. Let $\Phi$ \in. $\Phi$_{GY}^{v}. $\Phi$^{A}(t)=\displaystyle \int_{A} $\Phi$(x, t)d $\mu$(x). .. .. For. a. for all. (x, t)\in $\Omega$\times(0, \infty). subset A \subset $\Omega$ with 0. Then $\Phi$^{A}\in $\Phi$_{GY}.. < $\mu$(A). .. < \infty ,. let.
(7) 86. (i) \foral $\Phi$\in \mathcal{Y}^{(3)} \exists $\Psi$\in \mathcal{Y}^{(2)} s.t. $\Phi$\approx $\Psi$. (ii) \exists $\Phi$ \in $\Phi$_{Y}^{v} with $\Phi$(x, \cdot) \in y^{(1)} for each x but $\Phi$^{A} \in y^{(3)} Actually, let $\Omega$ (0,1) \subset \mathbb{R} with the Lebesgue measure and take Young functions 1 and $\Phi$(x, 1+x) $\Phi$(x, \cdot) \in y^{(1)} for all x \in $\Omega$ such that $\Phi$(x, 1) 2/x. Remark 2.1.. .. ,. =. =. Then. =. $\Phi$^{ $\Omega$}\in \mathcal{Y}^{(3)}.. Definition 2.5. Let. \overline{ $\Phi$}_{Y}, \overline{ $\Phi$}_{Y}^{v}, \overline{ $\Phi$}_{GY}. that $\Phi$\approx $\Psi$ for. $\Psi$ in. some. Definition 2.6. For. ,. \overline{ $\Phi$}_{GY}^{v}. ,. let. \'{I}_{ $\Omega$} $\Phi$(x, k|f(x)|)d $\mu$(x)<\infty { \displaystyle \Vert f\Vert_{L^{ $\Phi$} =\inf\{ $\lambda$>0 : \int_{ $\Omega$} $\Phi$(x, \frac{|f(x)|}{ $\lambda$})d $\mu$(x)\leq 1\}.. L^{ $\Phi$}( $\Omega$)= f\in L^{0}( $\Omega$). Example. 2.1. Let p=p. function defined we. denote. Example defined. on. with finite. for. :. L^{ $\Phi$}( $\Omega$)=L^{ $\Psi$}( $\Omega$). If $\Phi$\approx $\Psi$ , then. on. be. w. be. $\Omega$ valued in measure.. a. $\Omega$ valued in. L^{ $\Phi$}( $\Omega$) by L^{p(\cdot)}( $\Omega$) 2.2. Let. a. with. case we. variable exponent, that is, it is. (0, \infty ],. Example. and let. weight function,. (0, \infty). a.e., and a. a. },. $\Phi$(x, t) =t^{p(x)}. a. .. measurable In this. case. that is, it is. \displaystyle \int_{A}w(x)d $\mu$(x). a. measurable function. < \infty. for any A\subset $\Omega$. variable exponent, and let. L^{ $\Phi$}( $\Omega$) by L_{w}^{p(\cdot)}( $\Omega$). 2.3. Let p be. k>0. .. Let p be. denote. some. equivalent quasi‐norms.. $\Phi$(x, t)=t^{p(aj)}w(x) In this. \overline{$\Phi$} such. be the sets of all $\Phi$\in. $\Phi$_{Y}, $\Phi$_{Y}^{v}, $\Phi$_{G\mathrm{Y} and $\Phi$_{GY}^{v} respectively.. function $\Phi$\in. a. \overline{ $\Phi$}_{GY}^{v}. and. .. .. variable exponent, and let. $\Phi$(x,t)=\left\{ begin{ar ay}{l} 1/\exp(1/t^{p(x)} ,&t\in[0,1],\ \exp(t^{p(x)} ,&t\in(1,\infty]. \end{ar ay}\right. In this. case we. Next of ONeil. we. denote. recall the. [20,. L^{ $\Phi$}( $\Omega$) by \exp(IP^{(\cdot)})( $\Omega$) generalized. Definition. 1.2].. For. inverse of a. Young. .. Young function. $\Phi$ in the. function $\Phi$ and u\in. $\Phi$^{-1}(u)=\displaystyle \inf\{t\geq 0 : $\Phi$(t)>u\}. ,. [0, \infty]. ,. sense. let. (2.2).
(8) 87. where \displaystyle \inf\emptyset=\infty. respect. to t. For $\Phi$\in. .. by (2.2) for. \overline{ $\Phi$}_{GY}^{v}. each. x. ,. we. define also its. and denote it. inverse with. generalized. by $\Phi$^{-1}. .. That is,. $\Phi$^{-1}(x, u)=\displaystyle \inf\{t\geq 0 : $\Phi$(x,t)>u\}, (x, u)\in $\Omega$\times[0, \infty] \overline{ $\Phi$}_{GY}^{v},. Theorem 2.2. Let $\Phi$_{i}\in. i=1 ,. 2, 3. Assume that there. (2.3). .. exists. a. constant. C>0 such that. \displaystyle \frac{1}{C}$\Phi$_{2}^{-1}(x, t)\leq$\Phi$_{1}^{-1}(x, t)$\Phi$_{3}^{-1}(x,t)\leq C$\Phi$_{2}^{-1}(x,t). for. Assume also that there exists $\Psi$_{3}\in. for. some. \ell\in. (0,1 ]. \displaystyle \int_{A}$\Psi$_{3}(x, t)d $\mu$(x). .. and. for. (2.4). .. such that. $\Psi$_{3}^{A}( \cdot)^{1/\ell})\in \mathcal{Y}^{(1)}\cup y^{(2)}. and. $\Phi$_{3}\approx$\Psi$_{3}. $\Phi$_{GY}^{v}. (x, t)\in $\Omega$\times(0, \infty). any A\subset $\Omega$ with. 0< $\mu$(A). (2.5). ,. <\infty ,. where. $\Psi$_{3}^{A}(t)=. Then. \mathrm{P}\mathrm{W}\mathrm{M}(L^{$\Phi$_{1} ( $\Omega$), L^{$\Phi$_{2} ( $\Omega$))=L^{$\Phi$_{3} ( $\Omega$). ,. \Vert g\Vert_{\mathrm{o}\mathrm{p} \sim\Vert g\Vert_{L^{$\Phi$_{3} ( $\Omega$)}. Let p_{i} be variable exponents, w_{i} be. weight functions,. i=1 ,. 2, 3, and. $\Omega$_{\infty}=\{x\in $\Omega$:p_{3}(x)=\infty\}. Assume that. Example. \displaystyle \inf_{x\in $\Omega$}p_{i}(x)>0, i=1. ,. 2, 3, and. \displaystyle \sup_{x\in $\Omega$\backslash $\Omega$_{\infty} p_{3}(x)<\infty.. 2.4. Let. \displaystyle \frac{1}{p_{1}(x)}+\frac{1}{p_{3}(x)}=\frac{1}{p_{2}(x)}.. Then PWM. (L^{p_{1(\cdot)} ( $\Omega$), L^{p_{2(\cdot)} ( $\Omega$))=\ovalbox{\t \small REJECT} 3 (\cdot)( $\Omega$). ,. PWM (\exp(L^{p_{1(\cdot)}})( $\Omega$),\exp(L^{p_{2(\cdot)}})( $\Omega$))=\exp(L^{p_{3(\cdot)}})( $\Omega$). Example. .. 2.5. Let. \displaystyle \frac{1}{p_{1}(x)}+\frac{1}{p_{3}(x)}=\frac{1}{p_{2}(x)}, w_{1}(x)^{1/p_{1}(x)}w_{3}(x)^{1/p_{3}(x)}=w_{2}(x)^{1/p_{2}(x)}. Then. \mathrm{P}\mathrm{W}\mathrm{M} (\ovalbox{\t \smal REJECT} w11(\cdot)( $\Omega$), L_{w_{2}^{2(\cdot)} ^{p}( $\Omega$) =L_{w^{3}3}^{p(\cdot)}( $\Omega$). ..
(9) 88. Musielak‐Orlicz‐Morrey. 3. Let \mathbb{R}^{n} be the For. a. n ‐dimensional. function $\phi$. $\phi$(B)= $\phi$(x, r). :. \mathbb{R}^{n}\times. spaces. Euclidean space and $\mu$ the Lebesgue. (0, \infty). (0, \infty). \rightarrow. and. ball B=. a. B(x, r). ,. measure. we. write. .. Definition 3.1. (Musielak‐Orlicz‐Morrey space).. (0, \infty)\rightarrow(0, \infty). and. a. For $\Phi$ \in. \overline{ $\Phi$}_{GY}^{v},. $\phi$. :. \mathbb{R}^{n}. \times. ball B , let. \displaystyle \Vert f\Vert_{ $\Phi,\ \phi$,B}=\inf\{ $\lambda$>0 : \frac{1}{ $\phi$(B) $\mu$(B)}\int_{B} $\Phi$(x, \frac{|f(x)|}{ $\lambda$})d $\mu$(x)\leq 1\}, and let. L^{( $\Phi,\ \phi$)}(\mathbb{R}^{n})=\{f\in L^{0}(\mathbb{R}^{n}):\Vert f\Vert_{L( $\Phi,\ \phi$)}(\mathbb{R}^{n})<\infty\},. \displaystyle \Vert f\Vert_{L(\mathrm{I}\mathrm{R}^{n}) ( $\Phi,\ \phi$)=\sup_{B}\Vert f\Vert_{ $\Phi,\ \phi$,B}, where the supremum is taken If. $\phi$(B)=1/ $\mu$(B). For functions a. positive. $\theta$,. then. ,. $\kappa$ :. over. L^{( $\Phi,\ \phi$)}(\mathbb{R}^{n})=L^{ $\Phi$}(\mathbb{R}^{n}). \mathbb{R}^{n}\times(0, \infty)\rightarrow(0, \infty). \approx. .. we. ,. write $\theta$\sim $\kappa$ if there exists. constant C. such that. \displaystyle\frac{1}{C}\leq\frac{$\theta$(x,r)}{$\kap a$(x,r)}\leqC If $\Phi$. all balls B.. $\Psi$ and. $\phi$\sim $\psi$. ,. then. for all. L^{( $\Phi,\ \phi$)}(\mathbb{R}^{n}). (x, r)\in \mathbb{R}^{n}\times(0, \infty). =. L^{( $\Psi,\ \psi$)}(\mathbb{R}^{n}). with. .. equivalent quasi‐. norms.. Definition 3.2. A function $\theta$. (almost decreasing) a. positive. :. \mathbb{R}^{n}. \times. (0, \infty). with respect to the order. \rightarrow. by. (0, \infty). is almost. increasing. ball inclusion if there exists. constant C such that. $\theta$(B_{1})\leq C $\theta$(B_{2}). ( $\theta$(B_{1})\geq C $\theta$(B_{2})) for all balls B_{1} and B_{2} with B_{1}\subset B_{2}.. Definition 3.3. Let \mathcal{G}^{v} be the set of all that $\phi$ is almost. $\phi$(B) $\mu$(B). decreasing. is almost. $\phi$. :. \mathbb{R}^{n}. \times. (0, \infty). with respect to the order. increasing. \rightarrow. by ball. with respect to the order. (0, \infty). such. inclusion and. by ball inclusion..
(10) 89. \overline{ $\Phi$}_{GY}^{v}. Theorem 3.1. Let $\Phi$_{i} \in exists. $\phi$_{i} \in \mathcal{G}^{v},. 2, 3. Assume that there. i= 1 ,. constant C such that. positive. a. and. C^{-1}$\Phi$_{2}^{-1}(x, t$\phi$_{2}(x, r))\leq$\Phi$_{1}^{-1}(x, t$\phi$_{1}(x,r))$\Phi$_{3}^{-1}(x, t$\phi$_{3}(x, r)) for all x\in \mathbb{R}^{n} and r,t\in(0, \infty) \leq C$\Phi$_{2}^{-1}(x, t$\phi$_{2}(x, r and that. $\phi$_{3}/$\phi$_{1}. Assume also. is almost. one. (i) $\Phi$_{3} satisfies. (ii). increasing with respect. and. r,. the. ball inclusion.. $\Delta$_{2} condition, that is, $\Phi$_{3}(x, 2t)\leq\exists C_{$\Phi$_{3}}$\Phi$_{3}(x, t). $\phi$_{3}(x, r). is continuous with. .. respect. to. and, for all balls B,. (a) \exists$\Psi$_{B}\in y^{(1)} (b). by. of the following:. \displaystyle \lim_{r\rightar ow\infty}\inf_{x\in \mathbb{R}^{n} $\phi$_{3}(x,r) $\mu$(B(x, r) =\infty_{f} x. to the order. ,. \displaystyle \sup_{x\in B}$\Phi$_{3}(x,t)\leq$\Psi$_{B}(t) for. s.t.. all t,. and_{f}. \displaystyle \lim_{r\rightar ow+0}\inf_{x\in B}$\phi$_{3}(x, r)=\infty.. Then. \mathrm{P}\mathrm{W}\mathrm{M}(L^{($\Phi$_{1},$\phi$_{1})}(\mathbb{R}^{n}), L^{($\Phi$_{2},$\phi$_{2})}(\mathbb{R}^{n}) =L^{($\Phi$_{3},$\phi$_{3})}(\mathbb{R}^{n}). ,. \Vert g\Vert_{\mathrm{o}\mathrm{p} \sim\Vert g\Vert_{L^{($\Phi$_{3},$\phi$_{3})}(\mathb {R}^{n})}. Corollary w_{i} be. 3.2. Let p_{i}. weights. and. be variable exponents with. $\phi$_{i}\in \mathcal{G}_{y}^{v}i=1 2, ,. 1/p_{1}(x)+1/p_{3}(x)=1/p_{2}(x) that there exists. a. positive. 0<(p_{i})_{-}\leq(p_{i})_{+}\leq\infty,. 3. Assume that. ,. constant C such that. C^{-1}($\phi$_{2}(x, r)/w_{2}(x))^{1/\mathrm{p}_{2}(x)} \leq($\phi$_{1}(x,r)/w_{1}(x))^{1/p_{1}(x)}($\phi$_{3}(x, r)/w_{3}(x))^{1/p_{3}(x)} \leq C($\phi$_{2}(x, r)/w_{2}(x))^{1/p_{2}(x)}, for. $\phi$_{3}/$\phi$_{1} is almost increasing If (p_{3})_{+}<\infty then and that. all x\in \mathbb{R}^{n} and. with respect to the order. by. ,. PWM. (L_{w^{1} ^{(p_{1},$\phi$_{1})}(\mathbb{R}^{n}), L_{w_{2} ^{(p_{2},$\phi$_{2})}(\mathbb{R}^{n}) =L_{w_{3}^{3} ^{(p,$\phi$_{3})}(\mathbb{R}^{n}). \Vert g\Vert_{\mathrm{o}\mathrm{p} \sim\Vert g\Vert_{L_{w_{3} ^{(p_{3},$\phi$_{3}) (\mathb {R}^{n}) .. ,. r\in(0, \infty). ,. ball inclusion..
(11) 90. Corollary. 3.3. Let p_{i}. and $\lambda$_{i}. be variable exponents with 0. (p_{i})_{+}\leq\infty and-n\leq($\lambda$_{i})_{-}\leq($\lambda$_{i})_{+}<0, a. constant. w_{i} be. weightsf. <. (p_{i})_{-}. \leq. 2, 3. Let $\lambda$^{*} be. i=1 ,. with-n\leq$\lambda$^{*}<0 and let ,. $\phi$_{i}(x,r)=\left\{ begin{ar y}{l r^{$\lambda$_{i}(x)},&r\leq1/e,\ r^{$\lambda$^{*} &r>1/e. \end{ar y}\right. Assume that. (p_{3})_{+}. <. that $\lambda$_{i}. \infty ,. i. =. 1,. 2, 3_{f}. are. log‐Hölder continuous,. and that. \left{bginary}{l \frac{1}p_ (x)}+\frac{1}p_3(x)}=\frac{1}p_2(x)},\frac{$lmbda$_{1}(x)p_{1}(x)+\frac{$lmbda$_{3}(x)p_{3}(x)=\frac{$lmbda$_{2}(x)p_{2}(x),\ w_{1}(x)^/p_{1}(x)w_{3}(x)^1/p_{3}(x)=w_{2}(x)^1/p_{2}(x),\ $lambd$_{3}(x)\geq$lambd$_{1}(x),foralx\inmathb{R}^n. \ed{ary}\ight.. Then. \mathrm{P}\mathrm{W}\mathrm{M}(L_{w1}^{(p_{1},$\phi$_{1})}(\mathbb{R}^{n}), L_{w2}^{(p_{2},$\phi$_{2})}(\mathbb{R}^{n}) =L_{w}^{(p_{3},$\phi$_{3})}3(\mathbb{R}^{n}). ,. \Vert g\Vert_{\mathrm{o}\mathrm{p} \sim\Vert g\Vert_{L_{w_{3} ^{(p_{3},$\phi$_{3}) (\mathb {R}^{n}) . Corollary. 3.4. Let p_{i}. be variable exponents with. 0<(p_{i})_{-}\leq(p_{i})_{+}\leq\infty,. and let. $\Phi$_{i}(x,t)=\left\{ begin{ar ay}{l} 1/\exp(1/t^{p_{i}(x)} ,&t\in[0,1],\ \exp(t^{p_{i}(x)} ,&t\in(1,\infty], \end{ar ay}\right. Let $\lambda$ be. a. (p_{3})_{+}<\infty. constant with-1 < $\lambda$<0 , and let. and that. i=1 ,. 2, 3.. $\phi$(B)= $\mu$(B)^{ $\lambda$}. 1/p_{1}(x)+1/p_{3}(x)=1/p_{2}(x). .. .. Assume that. Then. \mathrm{P}\mathrm{W}\mathrm{M}(L^{($\Phi$_{1}, $\phi$)}(\mathbb{R}^{n})\cdot, L^{($\Phi$_{2}, $\phi$)}(\mathbb{R}^{n}) =L^{($\Phi$_{3}, $\phi$)}(\mathbb{R}^{n}). ,. \Vert g\Vert_{\mathrm{o}\mathrm{p} \sim\Vert g\Vert_{L(\mathb {R}^{n})}($\Phi$_{3},$\phi$_{\mathrm{J} ). The results in this section spaces defined. non‐doubling. on. spaces of. measure.. can. be extended to. homogeneous type. or. Musielak‐Orlicz‐Morrey. metric. measure. spaces with.
(12) 91. References [1]. L.. Diening,. Sobolev. P.. Harjulehto,. Spaces. P. Hästö and M.. Ruzicka, Lebesgue and. with variable exponents, Lecture Notes in Math. 2017,. 2011.. [2]. L. V. Kantorovich and G. P.. from the Russian. Akilov,. Functional. analysis.. Translated. by Howard L. Silcock. Second edition. Pergamon Press,. Oxford‐Elmsford, N.Y.,. 1982. \mathrm{x}\mathrm{i}\mathrm{v}+589 pp. ISBN:. 0‐08‐023036‐9; 0‐08‐. 026486‐7. [3]. P.. Kolwicz,. K. Leśnik and L.. Maligranda,. multipliers of. Pointwise. Calderón‐Lozanovskiĭ spaces, Math. Nachr. 286,. (2013),. 876‐. Maligranda, Pointwise products of. some. no.. 8‐9. 907.. [4]. P.. Kolwicz,. K. Leśnik and L.. Banach function spaces and factorization. J. Funct. Anal. 266 no.. (2014),. 2, 616‐659.. [5]. H.. Komatsu,. (Japanese) [Fourier analysis] Second edi‐ tion. Iwanami Shoten Kiso Sugaku [Iwanami Lectures on Fundamental Mathematics], 13. Kaisekigaku (I) [Analysis (I)], vi. Iwanami Shoten, Tokyo, 1983. \mathrm{v}\mathrm{i}+195 pp. ( $\Gamma$ irst edition is published in 1978 in Japanese). [6]. L.. Maligranda, Orlicz. ics. Fourier kaiseki.. spaces and. interpolation, Seminars. in mathemat‐. 5, Departamento de Matemática, Universidade Estadual de Camp‐. inas, Brasil, 1989.. [7]. L.. Maligranda and E. Nakai,. Math. 95. [8]. L.. (2010),. Maligranda. [9]. multipliers of Orlicz spaces, Arch.. 251‐256.. and L. E.. function spaces,. Pointwise. Indag.. Persson, Generalized duality of. Math. 51. (1989),. no.. 3,. some. Banach. 323‐338.. C. B. Morrey, Jr, On the solutions of quasi‐linear elliptic partial differ‐ ential equations, Trans. Amer. Math. Soc. 43. (1938),. no.. 1, 126‐166..
(13) 92. [10]. J.. Musielak, Orlicz. spaces and modular spaces, Lecture Notes in Math.. 1034, 1983. ISBN: 978‐3‐540‐12706‐2. [11]. E.. Nakai,. nology,. [12]. E.. Pointwise. (1995),. 37. (Print). E.. 85‐94.. Nakai, Pointwise multipliers. E.. Yokohama, 2004, E.. E.. Appl. Sci.. Morrey. 46. spaces. Mem. Osaka. (1997),. no.. (2000),. 1,. Ky‐. 1‐11.. on. the. Morrey. 3, 445‐454.. no.. on. Orlicz‐Morrey. Spaces (Kitakyushu, 2003),. Yokohama. spaces, Ba‐. Publishers,. 323‐333.. Nakai, The Campanato, Morrey and Hölder. mogeneous. [16]. the. Nakai, Generalized fractional integrals. nach and Function. [15]. on. Nakai, A characterization of pointwise multipliers. spaces, Sci. Math. 3. [14]. (Online). multipliers, Memoirs of The Akashi College of Tech‐. oiku Univ. III Natur. Sci.. [13]. 978‐3‐540‐38692‐6. type, Studia Math. 176. Nakai, Orlicz‐Morrey. (2006),. spaces and the. no.. spaces. on. spaces of ho‐. 1, 1‐19.. Hardy‐Littlewood maximal func‐. tion, Studia Mathematica, 188 (2008), 193‐221.. [17]. E.. Nakai, Pointwise multipliers. Math.. [18]. E.. J.,. on. Musielak‐Orlicz spaces, Nihonkai. to appear.. Nakai,. Pointwise. multipliers. on. Musielak‐Orlicz‐Morrey. spaces,. preparation.. [19]. E.. Nakai,. Pointwise. multipliers. on. several function spaces—a survey−,. preparation.. [20]. R.. ONeil,. Soc. 115. [21]. W.. Fractional. (1965),. Orlicz, Über. integration. in Orlicz spaces.. I., Trans. Amer. Math.. 300‐328. eine. Acad. Polonaise A. gewisse Klasse. von. Räumen. (1932), 207−220; reprinted. PWN, Warszawa 1988, 217‐230.. vom. Typus B, Bull.. in his Collected. Papers,.
(14) 93. [22]. W.. Orlicz, Über Räume. reprinted. [23]. [24]. in his Collected. M. M. Rao and Z. D.. New K.. (L^{M}). York, Basel. Yosida,. Heidelberg,. and. Bull. Acad. Polonaise A. Papers, PWN,. Warszawa. (1936), 93−107;. 1988, 345‐359.. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc.,. Hong Kong,. Fanctional New. ,. analysis,. York, Tokyo,. 1991.. sixth. edition, Springer‐Verlag, Berlin,. 1980.. Eiichi Nakai. Department of Mathematics University Mito, Ibaraki 310‐8512, Japan E‐‐mail address: [email protected] Ibaraki. \prime \mathfrak{N}+\mapsto \mathfrak{M}\star^{\backslash }\mathrm{F}^{\backslash }\mathfrak{B}\backslash \backslash \not\in_{\mathrm{D} ^{R} $\beta$ \#\#\overline{9\not\in}-.
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