Anisotropic Fractional Sobolev Spaces And Its Properties

The mixed-norm Lebesgue spaces can be treated as a natural generalization of the classical Lebesgue space.It derived from the study of H(?)rmander in translationinvariant operators in 1960.Moreover,in 1961,mixed-norm Lebesgue spaces were investigated systematically by Benedek and Panzone via using exponent vector p in place of the constant exponent p.Motivated by the aforementioned work of Benedek and Panzone,numerous other function spaces with mixed norms were gradually introduced and studied such as Besov spaces,Sobolev spaces,Bessel potential spaces,Triebel-Lizorkin spaces,Lorentz spaces,Banach function spaces,anisotropic Hardy spaces and Morrey spaces with mixed-norms.Meanwhile,classical fractional Sobolev spaces are based on the characterization of Gagliardo’s fractional derivatives.Compared to exact definition of derivatives,these sort of fractional derivatives are used to define the semi-norm with respect to these fractional Sobolev spaces in terms of integrals.In some practical applications,the integral representation in the global sense is able to research by some mathematical tools like Fourier transformation.Hence,fractional Sobolev spaces can be also used in several partial differential equations related to some special physical backgrounds that are contained both spacial and time variables.Since mixed-norm spaces have better space structures than classical function spaces to some extent,in this paper we mainly concentrate on the fractional Sobolev spaces in the sense of this mixed-norm,that is,anisotropic fractional Sobolev spaces.Consequently,we will construct our main results in this paper as follows.Fisrt of all,after we lead in the context of mixed-norm Lp spaces,mixed-norm inequalities and standard smooth moliffiers,we define anisotropic fractional Sobolev spaces Ws,p(Rn).At the same time,We demonstrate several properties with respect to this space simultaneously.When parameters s,s satisfy the order s<s,Ws,p(Rn)can be continuously embedded into Ws,p(Rn)in the sense of norm topology.W1,p(Rn)can be continuously embedded into any Ws,p(Rn).In addition,we prove that the pointwise porduct of a smooth function and an element of anisotropic fractional Sobolev spaces is again a function in itself.Based on this consequence,we can also obtain that Cc∞(Rn)is dense in Ws,p(Rn).For domains that shared special geometrical properties,we calculate several mixed-norm Poincare inequalities on these domains.Next,we prove the embedding theorem of anisotropic Sobolev spaces,that is,Ws,p(Rn)can be continuously embedded into Lq(Rn)under suitable parameter conditions.Specifically,on the one hand,since it is difficult to exchange the order of integral in mixed-norm calculations,and we cannot generalize series and integral estimates related to classical fractional Sobolev embedding results to mixed-norm case as well.Hence,in this paper we extend each element of anisotropic fractional Sobolev spaces into the domain R+× Rn.We also apply the Laplacian fundamental solution and convolutions to decompose this extended function into finite terms.On the other hand,compared to classical order cases between the prodcut sp and dimensional number n,we choose the arithmetic average Σi=1n si/n of s to define the critical vector p*=(np1/(n-s1p1),...,npn/(n-snpn)).We put relavant classical parameter conditions into each dimension of vector q by means of this critical vector p*.We also collect all vectors q that relied on vector p*that associated with different arithmetic combinatorial vector(s1,...,sn)to build parameters conditions for our embedding theorem.In the rest of this paper we delve the domian extension property of anisotropic fractional Sobolev spaces.In conclusion,we prove that each regular domain in Rn has an(s,p)-extension property.Particularly,during the verification process of extension property,we apply the Fefferman-Stein inequality of maximal functions to prove the mixed-morn boundedness of directional maximal function.we select a directional maximal function with special angles to estimate the Ws,p(Rn)-semi-norm form of general maximal functions pointwisely as well.Finally,we exploit the(s,p)-extension property to study the Holder regular property of elements in anisotropic fractional Sobolev spaces.We prove that for each regular domain Ω,Ws,p(Ω)can be continuously embedded in H(?)lder space C0.α(Ω),where α=s-1/p1-…-1/pn.