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Register a new userReverse Triangle Inequalities for Riesz Potentials and Connections with Polarization
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We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main tool used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure.
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We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequalities for potentials, and give applications of our results to the generalized polynomials. We also obtain sharp inequalities for products of norms of the weighted polynomials $w^nP_n, deg(P_n)le n,$ and for sums of suprema of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthest-point distance function.
For $sgeqslant d$, we obtain the leading term as $Nto infty$ of the maximal weighted $N$-point Riesz $s$-polarization (or Chebyshev constant) for a certain class of $d$-rectifiable compact subsets of $mathbb{R}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has $mathcal{H}_d$-measure zero, as well as finite unions of such sets when their pairwise intersections have $mathcal{H}_d$-measure zero. We also explicitly find the weak$^*$ limit distribution of asymptotically optimal $N$-point polarization configurations as $Nto infty$.
Let ${cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := {z in mathbb{C}: |z| leq 1, , , Im(z) geq 0}$$ be the closed upper half-disk of the complex plane. For integers $0 leq k leq n$ let ${mathcal F}_{n,k}^c$ be the set of all polynomials $P in {mathcal P}_n^c$ having at least $n-k$ zeros in $D^+$. Let $$|f|_A := sup_{z in A}{|f(z)|}$$ for complex-valued functions defined on $A subset {Bbb C}$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac{n}{k+1}right)^{1/2} leq inf_{P}{frac{|P^{prime}|_{[-1,1]}}{|P|_{[-1,1]}}} leq c_2 left(frac{n}{k+1}right)^{1/2}$$ for all integers $0 leq k leq n$, where the infimum is taken for all $0 otequiv P in {mathcal F}_{n,k}^c$ having at least one zero in $[-1,1]$. This is an essentially sharp reverse Markov-type inequality for the classes ${mathcal F}_{n,k}^c$ extending earlier results of Turan and Komarov from the case $k=0$ to the cases $0 leq k leq n$.
Let ${mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$|f|_A := sup_{x in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A subset {Bbb R}$. Let $$V_a^b(f) := int_a^b{|f^{prime}(x)| , dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 frac nkleq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{|P(1)|}} leq c_2 left( frac nk + 1 right)$$ for all integers $n geq 1$ and $k geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac nkright)^{1/2} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{|P(1)|}} leq c_2 left(frac nk + 1right)^{1/2}$$ for all integers $n geq 1$ and $k geq 1$.
We present reverse Holder inequalities for Muckenhoupt weights in $mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_infty$ weights with Fujii-Wilson constant $(w)_{A_infty}to 1^+$. That is, the local integrability exponent in the reverse Holder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Holder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Holder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Holder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.
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