The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces B_(p)^(s)(R^(n))=B_(p,p)^(s)(R^(n)),1≤p≤∞,and between Sobolev spaces Hs p(R^(n)),1<p<...The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces B_(p)^(s)(R^(n))=B_(p,p)^(s)(R^(n)),1≤p≤∞,and between Sobolev spaces Hs p(R^(n)),1<p<∞.In contrast to the paper H.Triebel,Mapping properties of Fourier transforms.Z.Anal.Anwend.41(2022),133–152,based mainly on embeddings between related weighted spaces,we rely on wavelet expansions,duality and interpolation of corresponding(unweighted)spaces,and(appropriately extended)Hausdorff-Young inequalities.The degree of compactness will be measured in terms of entropy numbers and approximation numbers,now using the symbiotic relationship to weighted spaces.展开更多
A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are ...A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.展开更多
In the recent years,the so-called Morrey smoothness spaces attracted a lot of interest.They can(also)be understood as generalisations of the classical spaces A_(p,q)^(s)(R^(n))with A∈{B,F}in R^(n),where the parameter...In the recent years,the so-called Morrey smoothness spaces attracted a lot of interest.They can(also)be understood as generalisations of the classical spaces A_(p,q)^(s)(R^(n))with A∈{B,F}in R^(n),where the parameters satisfy s∈R(smoothness),0<p∞(integrability)and 0<q∞(summability).In the case of Morrey smoothness spaces,additional parameters are involved.In our opinion,among the various approaches at least two scales enjoy special attention,also in view of applications:the scales A_(p,q)^(s)(R^(n))with A∈{N,E}and u≥p,and A_(p,q)^(s),τ(R^(n))with A∈{B,F}andτ≥0.We reorganise these two prominent types of Morrey smoothness spaces by adding to(s,p,q)the so-called slope parameter e,preferably(but not exclusively)with-n e<0.It comes out that|e|replaces n,and min(|e|,1)replaces 1 in slopes of(broken)lines in the(1/p,s)-diagram characterising distinguished properties of the spaces A_(p,q)^(s)(R^(n))and their Morrey counterparts.Special attention will be paid to low-slope spaces with-1<e<0,where the corresponding properties are quite often independent of n∈N.Our aim is two-fold.On the one hand,we reformulate some assertions already available in the literature(many of which are quite recent).On the other hand,we establish on this basis new properties,a few of which become visible only in the context of the offered new approach,governed,now,by the four parameters(s,p,q,e).展开更多
基金partially supported by the German Research Foundation(DFG)(Grant No.Ha 2794/8-1)。
文摘The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces B_(p)^(s)(R^(n))=B_(p,p)^(s)(R^(n)),1≤p≤∞,and between Sobolev spaces Hs p(R^(n)),1<p<∞.In contrast to the paper H.Triebel,Mapping properties of Fourier transforms.Z.Anal.Anwend.41(2022),133–152,based mainly on embeddings between related weighted spaces,we rely on wavelet expansions,duality and interpolation of corresponding(unweighted)spaces,and(appropriately extended)Hausdorff-Young inequalities.The degree of compactness will be measured in terms of entropy numbers and approximation numbers,now using the symbiotic relationship to weighted spaces.
文摘A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.
基金supported by the German Research Foundation (DFG) (Grant No.Ha2794/8-1)。
文摘In the recent years,the so-called Morrey smoothness spaces attracted a lot of interest.They can(also)be understood as generalisations of the classical spaces A_(p,q)^(s)(R^(n))with A∈{B,F}in R^(n),where the parameters satisfy s∈R(smoothness),0<p∞(integrability)and 0<q∞(summability).In the case of Morrey smoothness spaces,additional parameters are involved.In our opinion,among the various approaches at least two scales enjoy special attention,also in view of applications:the scales A_(p,q)^(s)(R^(n))with A∈{N,E}and u≥p,and A_(p,q)^(s),τ(R^(n))with A∈{B,F}andτ≥0.We reorganise these two prominent types of Morrey smoothness spaces by adding to(s,p,q)the so-called slope parameter e,preferably(but not exclusively)with-n e<0.It comes out that|e|replaces n,and min(|e|,1)replaces 1 in slopes of(broken)lines in the(1/p,s)-diagram characterising distinguished properties of the spaces A_(p,q)^(s)(R^(n))and their Morrey counterparts.Special attention will be paid to low-slope spaces with-1<e<0,where the corresponding properties are quite often independent of n∈N.Our aim is two-fold.On the one hand,we reformulate some assertions already available in the literature(many of which are quite recent).On the other hand,we establish on this basis new properties,a few of which become visible only in the context of the offered new approach,governed,now,by the four parameters(s,p,q,e).
