arXiv subjects
Publications and source records attributed to J. M. Aldaz.
Bombieri's inequality for polynomials requires that at least one of them be homogeneous. In this note an AI provides a natural example showing that, as stated in the title, there is no degree independent Bombieri type inequality for non-homogeneous polynomials.
The existence of decompositions of the form $f=P\cdot q+r$ with $P_k^{\ast}\left( D\right) r=0$, where $f$ is entire, $P$ a polynomial and $P^{\ast}_k$ the principal part of $P$ with its coefficients conjugated, was achieved in \cite{AlRe23} under certain restrictions on the order of $f$. Here we prove uniqueness, thereby obtaining Fischer decompositions, under conditions that sometimes match those required for existence, and sometimes are more restrictive, depending on the parameters involved.
We show that for all homogeneous polynomials $ f_{m}$ of degree $m$, in $d$ variables, and each $j = 1, \dots , d$, we have \begin{equation*} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}% ^{d-1}\right) } \geq \frac{π^{2}}{4\left( m+ 2 d + 1 \right)^{2}} \left \langle f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}^{d-1}\right) }. \end{equation*} This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem, when the data are given by entire functions of order sufficiently low on nonhyperbolic quadratic hypersurfaces.
We characterize the geometrically doubling condition of a metric space in terms of the uniform $L^1$-boundedness of superaveraging operators, where uniform refers to the existence of bounds independent of the measure being considered.
We explore boundedness properties in the context of metric measure spaces, of some natural operators of convolution type whose study is suggested by certain transformations used in computer vision.
We consider inequalities of Bombieri type for polynomials that need not be homogeneous, using the apolar inner product.
We continue the study initiated by H. S. Shapiro on Fischer decompositions of entire functions, showing that such decomposition exist in a weak sense (we do not prove uniqueness) under hypotheses regarding the order of the entire function $f$ to be expressed as $f= P\cdot q+r$, the polynomial $P$, and bounds on the apolar norm of homogeneous polynomials of degree $m$. These bounds, previously used by Khavinson and Shapiro, and by Ebenfelt and Shapiro, can be interpreted as a quantitative, asymptotic strengthening of Bombieri's inequality. In the special case where both the dimension of the space and the degree of $P$ are two, we characterize for which polynomials $P$ such bounds hold.
Let $P$ be a fixed homogeneous polynomial. We present a sharp condition on $P$ guaranteeing the existence of asymptotically larger bounds in Bombieri's inequality, so for every homogeneous polynomial $q_m$ of degree $m$ we have \begin{equation*} \left\Vert P q_{m}\right\Vert _{a}\geq C_{P} m^{l\left( P\right) /2}\left\Vert q_{m}\right\Vert _{a}, \end{equation*} where $\| \cdot \| _{a}$ denotes the apolar norm. Explicit estimates for $C_P > 0$ and $l(P) > 0$ are given.
Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there exist $C>0$, $D>0$ and $α\ge 0$ such that \begin{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left( m+D\right) ^{α}}\left\langle f_{m},f_{m}\right\rangle_{\mathbb{S}^{d-1}} \end{equation*} for all homogeneous polynomials $f_{m}$ of degree $m.$ Assume that $P_{j}$ for $j=0, \dots ,β<2k$ are homogeneous polynomials of degree $j$. The main result of the paper states that for any entire function $f$ of order $% ρ<\left( 2k-β\right) /α$ there exist entire functions $q$ and $h$ of order bounded by $ρ$ such that \begin{equation*} f=\left( P_{2k}-P_{β}- \dots -P_{0}\right) q+h\text{ and }Δ^{h}r=0. \end{equation*} This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem for parabola-shaped domains on the plane, with data given by entire functions of order smaller than $\frac{1}{2}$.
We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, and also consider weak versions of doubling, in connection to the non-uniqueness of centers and radii in arbitrary metric spaces.
We prove that in a metric measure space $X$, if for some $p \in (1,\infty)$ there are uniform bounds (independent of the measure) for the weak type $(p,p)$ of the centered maximal operator, then $X$ satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type $(1,1)$ of the centered operator. Thus, the following characterization is obtained: the centered maximal operator satisfies uniform weak type $(1,1)$ bounds if and only if the space $X$ has the Besicovitch intersection property. In $\mathbb{R}^d$ with any norm, the constants coming from the Besicovitch intersection property are bounded above by the translative kissing numbers. The extensive literature on kissing numbers allows us to obtain, first, sharp estimates on the uniform bounds satisfied by the centered maximal operators defined by arbitrary norms on the plane, second, sharp estimates in every dimension when the $\ell_\infty$ norm is used, and third, improved estimates in all dimensions when considering euclidean balls, as well as the sharp constant in dimension 3. Additionally, we prove that the existence of uniform $L^1$ bounds for the averaging operators associated to arbitrary measures and radii, is equivalent to a weaker variant of the Besicovitch intersection property.
We prove that in a metric measure space $(X, d, μ)$, the averaging operators $A_{r, μ}$ satisfy a uniform strong type $(1,1)$ bound $\sup_{r, μ} \|A_{r, μ}\|_{L^1\to L^1} < \infty$ if and only if $X$ satisfies a certain geometric condition, the equal radius Besicovitch intersection property.
We show that under rather general circumstances, the almost everywhere pointwise inequality $|f|(x) \le Mf (x)$ is equivalent to a weak form of the Lebesgue density theorem, for totally bounded closed sets. We derive both positive and negative results from this characterization.
We prove that averaging operators are uniformly bounded on $L^1$ for all geometrically doubling metric measure spaces, with bounds independent of the measure. From this result, the $L^1$ convergence of averages as $r \to 0$ immediately follows.
We study generalizations of the classical Bernstein operators on polynomial spaces, where instead of fixing $\mathbf{1}$ and $x$, we require that $\mathbf{1}$ and a strictly increasing polynomial $f_1$ be fixed. Via several examples, we exhibit the diversity of behaviours in this more general setting. We also prove that for sufficiently large dimensions, there always exist generalized Bernstein operators fixing $\mathbf{1}$ and $f_1$, and converging to the identity.
We explore some variants of the Boman covering lemma, and their relationship to the boundedness properties of the maximal operator. Let $1 < p < \infty$ and let $q$ be its conjugate exponent. We prove that the strong type $(q,q)$ of the uncentered maximal operator, by itself, implies certain generalizations of the Boman covering lemma for the exponent $p$, and in turn, these generalizations entail the weak type $(q,q)$ of the centered maximal operator. We show by example that it is possible for the uncentered maximal operator to be unbounded for all $1 < s < \infty$, while the conclusion of the lemma holds for every $1 < p < \infty$; thus, the latter condition is much weaker. Also, the boundedness of the centered maximal operator entails weak versions of the lemma.
We study certain generalizations of the classical Bernstein operators, defined via increasing sequences of nodes. Such operators are required to fix two functions, $f_0$ and $f_1$, such that $f_0 > 0$ and $f_1/ f_0$ is increasing on an interval $[a,b]$. A characterization regarding when this can be done is presented. From it we obtain, under rather general circumstances, the following necessary condition for existence: if nodes are non-«decreasing, then $(f_1/f_0)^\prime >0 $ on $(a,b)$, while if nodes are strictly increasing, then $(f_1/f_0)^\prime >0 $ on $[a,b]$.
In \cite{NaTa} Naor and Tao extended to the metric setting the $O(d \log d)$ bounds given by Stein and Strömberg for Lebesgue measure in $\mathbb{R}^d$, deriving these bounds first from a localization result, and second, from a random Vitali lemma. Here we show that the Stein-Strömberg original argument can also be adapted to the metric setting, giving a third proof. We also weaken the hypotheses, and additionally, we sharpen the estimates for Lebesgue measure.