In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\<p\<+\infty$ and $0\<s\<1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$...

We show that the differential inequality $\left|\Delta u\right|\le v\left|u\right|$ has the unique continuation property relative to the Sobolev space $H_{loc}^{2,1}\left(\Omega \right)$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition $$(K_n^{\mathrm{loc}})\underset{r\to 0}{lim}\underset{x\in K}{sup}{\int }_{|x-y|\<r}|x-y{|}^{2-n}v\left(y\right)dy=0$$ for all compact $K\subset \Omega$, where if $n=2$, we replace $|x-y{|}^{2-n}$ by $-log|x-y|$. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,...$ and any $u\in H_c^{2,1}({\mathbf{R}}^3-\left\{0\right\}),$ $$\frac{\left|u\right(x\left)\right|}{|x{|}^N}\le C{\int }_{{\mathbf{R}}^3}|x-y{|}^{-1}\frac{\left|\Delta u\right(y\left)\right|}{|y{|}^N}dy\text{for}\text{a.e.}x\text{in}{\mathbf{R}}^3.$$

We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f\in L_{2,-\mu }$, $v₀\in H_{-\mu }¹$, μ ∈ (0,1). We prove an estimate of the velocity in the $H_{-\mu }^{2,1}$ norm and of the gradient of the pressure in the norm of $L_{2,-\mu }$. We apply the Fourier transform with respect to the variable along...

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^m{p}_i\left(n\right)x\left({\tau }_i\left(n\right)\right)=0,n\ge 0\left[\nabla x\left(n\right)-\sum _{i=1}^m{p}_i\left(n\right)x\left({\sigma }_i\left(n\right)\right)=0,n\ge 1\right],$$ where $\left(p_i\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau }_i\left(n\right)$ [${\sigma }_i\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta$ [$\nabla$] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=\tau \left(n\right)}^n{p}_i\left(j\right)>1\left[\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=n}^{\sigma \left(n\right)}{p}_i\left(j\right)>1\right]$$ and $$\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j={\tau }_i\left(n\right)}^{n-1}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\left[\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j=n+1}^{{\sigma }_i\left(n\right)}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\right]$$ are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau }_i\left(n\right)$ $[\sigma \left(n\right)={min}_{1\le i\le m}{\sigma }_i\left(n\right)]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator $S_d$ and its associated global maximal operator $S*{*}_d$ by $\left(S_{d}f\right)(x,t)=1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S*{*}_{d}f)\left(x\right)=su{p}_{t\in ℝ}|1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi |$, f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, $S_{d}f$ is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

In this paper, characterizations of the embeddings between weighted Copson function spaces ${\mathrm{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\mathrm{Ces}}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$\begin{array}{cc}\hfill d({\int }_0^{\infty }& {\left({\int }_0^{t}f{\left(\tau \right)}^{p_2}{v}_2\left(\tau \right)\mathrm{d}\tau \right)}^{q_2/p_2}{u}_2\left(t\right)\mathrm{d}t{)}^{1/q_2}\hfill \\ & \le c{\left({\int }_0^{\infty }{\left({\int }_t^{\infty }f{\left(\tau \right)}^{p_1}{v}_1\left(\tau \right)\mathrm{d}\tau \right)}^{q_1/p_1}{u}_1\left(t\right)\mathrm{d}t\right)}^{1/q_1},\hfill \end{array}d$$ where $p_1,p_2,q_1,q_2\in (0,\infty )$, $p_2\le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques...

Let $u$ be a weight on $(0,\infty )$. Assume that $u$ is continuous on $(0,\infty )$. Let the operator $S_u$ be given at measurable non-negative function $\varphi$ on $(0,\infty )$ by $$S_u\varphi \left(t\right)=\underset{0<\tau \le t}{sup}u\left(\tau \right)\varphi \left(\tau \right).$$ We characterize weights $v,w$ on $(0,\infty )$ for which there exists a positive constant $C$ such that the inequality $${\left({\int }_0^{\infty }{\left[S_u\varphi \left(t\right)\right]}^{q}w\left(t\right)dt\right)}^{\frac{1}{q}}\lesssim {\left({\int }_0^{\infty }{\left[\varphi \left(t\right)\right]}^{p}v\left(t\right)dt\right)}^{\frac{1}{p}}$$ holds for every $0<p,q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.