We study embeddings between generalised Triebel–Lizorkin–Morrey spacesε_(ϕ,p,q)^(s)(R^(d))and within the scales of further generalised Morrey smoothness spaces like N_(ϕ,p,q)^(s)(R^(d)),B_(p,q)^(s,ϕ)(R^(d))and F_(p...We study embeddings between generalised Triebel–Lizorkin–Morrey spacesε_(ϕ,p,q)^(s)(R^(d))and within the scales of further generalised Morrey smoothness spaces like N_(ϕ,p,q)^(s)(R^(d)),B_(p,q)^(s,ϕ)(R^(d))and F_(p,q)^(s,ϕ)(R^(d)).The latter have been investigated in a recent paper by the first two authors(2023),while the embeddings of the scale N_(ϕ,p,q)^(s)(R^(d))were mainly obtained in a paper of the first and last two authors(2022).Now we concentrate on the characterisation of the spacesε_(ϕ,p,q)^(s)(R^(d)).Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies’wavelets.Then we prove necessary and sufficient conditions for the embeddingε_(ϕ1,p1,q1)^(s1)(R^(d))→ε_(2ϕ2,p2,q2)^(s)(R^(d)).We can also provide some almost final answer to the question whenε_(ϕ,p,q)^(s)(R^(d))is embedded into C(R^(d)),complementing our recent findings in case of N_(ϕ,p,q)^(s)(R^(d)).展开更多
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
For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its...For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.展开更多
In the last few decades,gravastars have been proposed as an alternative to black holes.The stability of a gravastar has been examined in many modified theories of gravity along with Einstein's GR.The f(Q,T)gravity...In the last few decades,gravastars have been proposed as an alternative to black holes.The stability of a gravastar has been examined in many modified theories of gravity along with Einstein's GR.The f(Q,T)gravity,a successfully modified theory of gravity for describing the current accelerated expansion of the universe,has been used in this study to examine gravastar in different aspects.According to Mazur and Mottola[Proc.Natl.Acad.Sci.101,9545(2004);Gravitational condensate stars:An alternative to black holes,I12-011,(2002)],a gravastar has three regions with three different equations of state.In this study,we examined the interior of a gravastar by consid-ering p=-ρ EoS to describe the dark sector for the interior region.The next region is a thin shell of ultrarelativistic stiff fluid,in which we investigated several physical properties,including proper length,energy,entropy,and surface energy density.Additionally,we examined the surface redshift and speed of sound to check the potential stability of our proposed thin-shell gravastar model.Furthermore,we used the entropy maximization technique to verify the stability of the gravastar model.A gravastar's outer region is a complete vacuum described by exterior Schwarzschild geometry.Finally,we presented a stable gravastar model,which is singularity-free and devoid of any incom-pleteness in classical black hole theory.展开更多
基金partially supported by the German Research Foundation(DFG)(Grant No.Ha 2794/8-1)supported by the China Scholarship Council(CSC)(Grant No.202006350058)partially supported by the Center for Mathematics of the University of Coimbra(funded by the Portuguese Government through FCT/MCTES,DOI 10.54499/UIDB/00324/2020)。
文摘We study embeddings between generalised Triebel–Lizorkin–Morrey spacesε_(ϕ,p,q)^(s)(R^(d))and within the scales of further generalised Morrey smoothness spaces like N_(ϕ,p,q)^(s)(R^(d)),B_(p,q)^(s,ϕ)(R^(d))and F_(p,q)^(s,ϕ)(R^(d)).The latter have been investigated in a recent paper by the first two authors(2023),while the embeddings of the scale N_(ϕ,p,q)^(s)(R^(d))were mainly obtained in a paper of the first and last two authors(2022).Now we concentrate on the characterisation of the spacesε_(ϕ,p,q)^(s)(R^(d)).Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies’wavelets.Then we prove necessary and sufficient conditions for the embeddingε_(ϕ1,p1,q1)^(s1)(R^(d))→ε_(2ϕ2,p2,q2)^(s)(R^(d)).We can also provide some almost final answer to the question whenε_(ϕ,p,q)^(s)(R^(d))is embedded into C(R^(d)),complementing our recent findings in case of N_(ϕ,p,q)^(s)(R^(d)).
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.
基金the National Natural Science Foundation of China (Grant Nob. 10426014, 10501010 and 10201004)Important Fund of Hubei Provincial Department of Education (Grant No.D200510005)
文摘For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.
基金SP&PKS acknowledges the National Board for Higher Mathematics(NBHM)under the Department of Atomic Energy(DAE),Govt.of India for financial support to carry out the Research project No.:02011/3/2022 NBHM(R.P.)/R&D II/2152 Dt.14.02.2022.PKS thanks Transilvania University of Brasov for Transilvania Fellowship for Visiting Professors。
文摘In the last few decades,gravastars have been proposed as an alternative to black holes.The stability of a gravastar has been examined in many modified theories of gravity along with Einstein's GR.The f(Q,T)gravity,a successfully modified theory of gravity for describing the current accelerated expansion of the universe,has been used in this study to examine gravastar in different aspects.According to Mazur and Mottola[Proc.Natl.Acad.Sci.101,9545(2004);Gravitational condensate stars:An alternative to black holes,I12-011,(2002)],a gravastar has three regions with three different equations of state.In this study,we examined the interior of a gravastar by consid-ering p=-ρ EoS to describe the dark sector for the interior region.The next region is a thin shell of ultrarelativistic stiff fluid,in which we investigated several physical properties,including proper length,energy,entropy,and surface energy density.Additionally,we examined the surface redshift and speed of sound to check the potential stability of our proposed thin-shell gravastar model.Furthermore,we used the entropy maximization technique to verify the stability of the gravastar model.A gravastar's outer region is a complete vacuum described by exterior Schwarzschild geometry.Finally,we presented a stable gravastar model,which is singularity-free and devoid of any incom-pleteness in classical black hole theory.
