Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. $L^p$, 1 ≤ p ≤ ∞) sense at $x\in {ℝ}^m$ if there are numbers $f_{\alpha }\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum }_{\left|\alpha \right|\le n}{f}_{\alpha }\left(x\right)h^{\alpha }/\alpha !$ is $O\left(h^u\right)$ in the approximate (resp. $L^p$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or $L^p$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...

Let $X_1,\cdots ,X_n$ be Banach spaces and $f$ a real function on $X=X_1\times \cdots \times X_n$. Let $A_f$ be the set of all points $x\in X$ at which $f$ is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if $X_1,\cdots ,X_{n-1}$ are Asplund spaces and $f$ is continuous (respectively Lipschitz) on $X$, then $A_f$ is a first category set (respectively a $\sigma$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X\to Y$ is a Lipschitz mapping, then there exists a $\sigma$-upper porous set $A\subset X$ such that $f$ is Fréchet differentiable...

In this note we shall prove that for a continuous function $\varphi :\Delta \to {ℝ}^n$, where $\Delta \subset ℝ$,  the paratingent of $\varphi$ at $a\in \Delta$ is a non-empty and compact set in ${ℝ}^n$ if and only if $\varphi$ satisfies Lipschitz condition in a neighbourhood of $a$. Moreover, in this case the paratingent is a connected set.

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(\alpha ₁,...,{\alpha }_N)$ for some $0<\alpha ₁,...,{\alpha }_N<1$ and its marginal functions satisfy $f(·,x₂,...,x_N)\in Lip\beta ₁,...,f(x₁,...,x_{N-1},·)\in Lip{\beta }_N$ for some $0<\beta ₁,...,{\beta }_N<1$ uniformly in the indicated variables $x_l$, 1 ≤ l ≤ N, then $f{̃}^{(\eta ₁,...,{\eta }_N)}\in Lip(\alpha ₁,...,{\alpha }_N)$ for each choice of $(\eta ₁,...,{\eta }_N)$ with ${\eta }_l=0$ or 1 for 1 ≤ l ≤ N.

Let $F$ be a closed subset of ${ℝ}^n$ and let $P\left(x\right)$ denote the metric projection (closest point mapping) of $x\in {ℝ}^n$ onto $F$ in $l^p$-norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in ${ℝ}^n$ in the Euclidean case $p=2$. We consider the case $2<p<\infty$ and prove that the $i$th component $P_i\left(x\right)$ of $P\left(x\right)$ is differentiable a.e. if $P_i\left(x\right)\ne x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i\left(x\right)=x_i$.

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new...

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\&lt;\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

We consider a double analytic family of fractional integrals $S_z^{\gamma ,\alpha }$ along the curve ${t\mapsto \left|t\right|}^{\alpha }$, introduced for α = 2 by L. Grafakos in 1993 and defined by $\left(S_z^{\gamma ,\alpha }f\right)(x₁,x₂):=1/\Gamma (z+1/2){\int \int |u-1|}^z\psi (u-1){f(x₁-t,x₂-u|t|}^{\alpha }{\left)du\right|t|}^{\gamma }dt/t$, where ψ is a bump function on ℝ supported near the origin, $f{\in }_c^{\infty }\left(ℝ²\right)$, z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_z^{\gamma ,\alpha }$ maps $L^p\left(ℝ²\right)$ to $L^q\left(ℝ²\right)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{-1+i\theta }^{i\varrho ,\alpha }$ is a product kernel on ℝ², adapted to the curve ${t\mapsto \left|t\right|}^{\alpha }$; as a consequence, we show...

It is known that there is no natural Banach norm on the space $\mathrm{\mathscr{H}𝒦}$ of $n$-dimensional Henstock-Kurzweil integrable functions on $[a,b]$. We show that the $\mathrm{\mathscr{H}𝒦}$ space is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. On each $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space, an $F$-norm ${\parallel ·\parallel }^X$ is defined. A ${\parallel ·\parallel }^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a ${\parallel ·\parallel }^X$-continuous linear operator. Hence, many important results in functional analysis hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is well-known that every...

We prove that for a normed linear space $X$, if $f_1:X\to ℝ$ is continuous and semiconvex with modulus $\omega$, $f_2:X\to ℝ$ is continuous and semiconcave with modulus $\omega$ and $f_1\le f_2$, then there exists $f\in C^{1,\omega }\left(X\right)$ such that $f_1\le f\le f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega \left(t\right)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C_{loc}^{1,\omega }$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

For 0 < γ ≤ 1, let $\Lambda {⁺}_{\gamma }$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{\gamma }(E,Q)$ be the closed ideal in $\Lambda {⁺}_{\gamma }$ consisting of those functions $f\in \Lambda {⁺}_{\gamma }$ for which (i) f = 0 on E, (ii) $|f\left(z\right)-f\left(w\right)|=o\left(\right|z-w{|}^{\gamma })$ as d(z,E),d(w,E) → 0, (iii) $f/Q\in \Lambda {⁺}_{\gamma }$. Also, for a closed ideal I in $\Lambda {⁺}_{\gamma }$, let $E_I$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_I$ be the greatest common divisor of the inner parts of non-zero functions...

Let E be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of ${ℂ}_{\infty }\setminus E$ with pole at infinity and $Mₙ\left(E\right)=sup\left(\right||P^{\text{'}}{\left|\right|}_E{)/(\left|\right|P\left|\right|}_E)$ where the supremum is taken over all polynomials ${P|}_E\not\equiv 0$ of degree at most n, and ${\left|\right|f\left|\right|}_E=sup\left|f\right(z\left)\right|:z\in E$. The paper deals with recent results concerning a connection between the smoothness of $g_E$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${Mₙ\left(E\right)}_{n=1,2,...}$. Some additional conditions are given for special classes of sets.

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.

We consider the integral functional $J\left(u\right)={\int }_{\Omega }\left[f\right(\left|Du\right|\left)-u\right]dx$, $u\in W_0^{1,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^n$, $n\ge 2$, is a nonempty bounded connected open subset of ${ℝ}^n$ with smooth boundary, and $ℝ\ni s\mapsto f\left(\right|s\left|\right)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_0^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

Inspired by a result from Leibov, we find that the supremum defining the $BLO$ norm in $[0,1]$ is actually attained by a specific sub-interval of $[0,1]$ for $f\in VLO([0,1])$