We study the inequalities of the type $|int_{mathbb{R}^d} Phi(K*f)| lesssim |f|_{L_1(mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $alpha - d$ and possibly vector-valued, the function $Phi$ is positively $p$-homogeneous, and $p = d/(
d-alpha)$. Under mild regularity assumptions on $K$ and $Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.
Answering a key point left open in a recent work of Bongers, Guo, Li and Wick, we provide the following lower bound $$ |b|_{text{BMO}_{gamma}(mathbb{R}^2)}lesssim |[b,H_{gamma}]|_{L^p(mathbb{R}^2)to L^p(mathbb{R}^2)}, $$ where $H_{gamma}$ is the parabolic Hilbert transform.
We study the rational solutions of the Abel equation $x=A(t)x^3+B(t)x^2$ where $A,Bin C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper bound on the nu
mber of rational solutions is obtained. Moreover, we prove that if there are more than $(deg(A)+1)/2$ rational solutions then the equation admits a Darboux first integral.
Recent work showed that a theorem of Joris (that a function $f$ is smooth if two coprime powers of $f$ are smooth) is valid in a wide variety of ultradifferentiable classes $mathcal C$. The core of the proof was essentially $1$-dimensional. In certai
n cases a multidimensional version resulted from subtle reduction arguments, but general validity, notably in the quasianalytic setting, remained open. In this paper we give a uniform proof which works in all cases and dimensions. It yields the result even on infinite dimensional Banach spaces and convenient vector spaces. We also consider more general nonlinear conditions, namely general analytic germs $Phi$ instead of the powers, and characterize when $Phi circ f in mathcal C$ implies $f in mathcal C$.
This paper concerns existence of right-continuous with bounded variation solutions of a perturbed second-order differential inclusion governed by time and state-dependent maximal monotone operators.
We consider the set of monic real univariate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities of their positive and negative roots (all roots are distinct). For $d=6$ an
d for signs of the coefficients $(+,-,+,+,+,-,+)$, we prove that the set of such polynomials having two positive, two negative and two complex conjugate roots, is not connected.
The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond, Math. Ann., 21(1):109-117, 1883). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky, Cre
lle, 80:1-32, 1875) that a solution to the heat equation may not be time-analytic at $t=0$ even if the initial function is real analytic. Recently, it was shown in cite{Zha20, DZ20, DP20} that solutions to the heat equation in the whole space, or half space with zero boundary value, are analytic in time under essentially optimal conditions. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any $delta>0$, we find a solution to the heat equation on the whole plane, with exponential growth of order $2+delta$, which is nowhere analytic in time.
In this paper, we provide both a preservation and breaking of symmetry theorem for $2pi$-periodic problems of the form begin{align*} begin{cases} -u(t) + g(u(t)) = f(t)cr u(0) - u(2pi) = u(0) - u(2pi) = 0 end{cases} end{align*} where $g: mathbb{R} to
mathbb{R}$ is a given $C^1$ function and $f: [0,2pi] to mathbb{R}$ is continuous. We provide a preservation of symmetry result that is analogous to one given by Willem (Willem, 1989) and a generalization of the theorem given by Costa-Fang (Costa and Fang, 2019). Both of these theorems use group actions that are not normally considered in the literature.
In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painleve equations using the geometric framework of Okamotos space of initial values. W
e demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in [HC17], we show how the recurrence coefficients are connected to the fourth Painleve equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in [FVA18] we discuss the relation of the recurrence coefficients to the sixth Painleve equation. In addition to demonstrating the general scheme, these results supplement previous studies [DFS20, HFC20], and we also discuss a number of related topics in the context of the geometric approach, such as Hamiltonian forms of the differential equations for the recurrence coefficients, Riccati solutions for special parameter values, and associated discrete Painleve equations.
We show that Sturms classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a unique turning point in the princi
pal part of the equation. Related results are discussed. The last section contains an extension of the main result to a finite number of turning points.
