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Register a new userBilinear Sobolev-Poincare inequalities and Leibniz-type rules
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The dual purpose of this article is to establish bilinear Poincare-type estimates associated to an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato-Morrey spaces under Sobolev scaling.
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We study weighted Poincare and Poincare-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ left (frac{1}{w(Q)}int_Q|f-f_Q|^{q}wright )^frac{1}{q}le C_well(Q)left (frac{1}{w(Q)}int_Q | abla f|^p wright )^frac{1}{p}, $$ with different quantitative estimates for both the exponent $q$ and the constant $C_w$. We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality $$ frac{1}{|Q|}int_Q |f-f_Q| dmu le a(Q), $$ for all cubes $Qsubsetmathbb{R}^n$ and where $a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev-type exponent $p^*_w>p$ associated to the weight $w$ and obtain further improvements involving $L^{p^*_w}$ norms on the left hand side of the inequality above. For the endpoint case of $A_1$ weights we reach the classical critical Sobolev exponent $p^*=frac{pn}{n-p}$ which is the largest possible and provide different type of quantitative estimates for $C_w$. We also show that this best possible estimate cannot hold with an exponent on the $A_1$ constant smaller than $1/p$. We also provide an argument based on extrapolation ideas showing that there is no $(p,p)$, $pgeq1$, Poincare inequality valid for the whole class of $RH_infty$ weights by showing their intimate connection with the failure of Poincare inequalities, $(p,p)$ in the range $0<p<1$.
In this paper we give a geometric condition which ensures that $(q,p)$-Poincare-Sobolev inequalities are implied from generalized $(1,1)$-Poincare inequalities related to $L^1$ norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several $(1,1)$-Poincare type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincare-Sobolev estimates. Among other results, we prove that for each rectangle $R$ of the form $R=I_1times I_2 subset mathbb{R}^{n}$ where $I_1subset mathbb{R}^{n_1}$ and $I_2subset mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have that % begin{equation*} left( frac{1}{w(R)}int_{ R } |f -f_{R}|^{p_{delta,w}^*} ,wdxright)^{frac{1}{p_{delta,w}^*}} leq c,delta^{frac1p}(1-delta)^{frac1p},[w]_{A_{1,mathfrak{R}}}^{frac1p}, Big(a_1(R)+a_2(R)Big), end{equation*} % where $delta in (0,1)$, $w in A_{1,mathfrak{R}}$, $frac{1}{p} -frac{1}{ p_{delta,w}^* }= frac{delta}{n} , frac{1}{1+log [w]_{A_{1,mathfrak{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{delta,p}(Q)}$ (See Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincare-Sobolev estimates with the gain $delta^{frac1p}(1-delta)^{frac1p}$.
In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product begin{equation*} leftlangle p,qrightrangle _{s}=int_{mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{prime }(0)q^{prime }(0), end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.
We study the two-weighted estimate [ bigg|sum_{k=0}^na_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg|leq c|f|L_{p,u}(0,infty)|,tag{$*$} ] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<pleq qleqinfty$, provided that the weight $u$ satisfies the certain conditions, the estimate $(*)$ holds if and only if the estimate [ sum_{k=0}^nbigg|a_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg| leq c|f|L_{p,u}(0,infty)|.tag{$**$} ] is fulfilled. The necessary and sufficient conditions for $(**)$ to be valid are well-known. The obtained result can be applied to the estimates of differential operators with variable coefficients in some weighted Sobolev spaces.
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
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