Analysis of Partial Differential Equations (APDE)http://hdl.handle.net/20.500.11824/12024-03-01T10:37:26Z2024-03-01T10:37:26ZControl of pseudodifferential operators by maximal functions via weighted inequalitiesBeltran, D.http://hdl.handle.net/20.500.11824/17442024-02-09T23:18:04Z2019-01-01T00:00:00ZControl of pseudodifferential operators by maximal functions via weighted inequalities
Beltran, D.
We establish general weighted L 2 inequalities for pseudodifferential operators associated to the Hörmander symbol classes S ρ,δm . Such inequalities allow one to control these operators by fractional “non-tangential” maximal functions and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt-type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar–Stein almost orthogonality principle and a quantitative version of the symbolic calculus.
2019-01-01T00:00:00ZSubdyadic square functions and applications to weighted harmonic analysisBeltran, D.Bennett, J.http://hdl.handle.net/20.500.11824/17432024-02-09T23:18:03Z2017-02-05T00:00:00ZSubdyadic square functions and applications to weighted harmonic analysis
Beltran, D.; Bennett, J.
Through the study of novel variants of the classical Littlewood–Paley–Stein g-functions, we obtain pointwise estimates for broad classes of highly-singular Fourier multipliers on Rd satisfying regularity hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to efficiently bound such multipliers by geometrically-defined maximal operators via general weighted L2 inequalities, in the spirit of a well-known conjecture of Stein. Our framework applies to solution operators for dispersive PDE, such as the time-dependent free Schrödinger equation, and other highly oscillatory convolution operators that fall well beyond the scope of the Calderón–Zygmund theory.
2017-02-05T00:00:00ZA Fefferman-Stein inequality for the Carleson operatorBeltran, D.http://hdl.handle.net/20.500.11824/17422024-02-09T23:18:01Z2018-01-01T00:00:00ZA Fefferman-Stein inequality for the Carleson operator
Beltran, D.
We provide a Fefferman-Stein type weighted inequality for maximally modulated Calderón-Zygmund operators that satisfy a priori weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of Pérez. Applying it to the Hilbert transform we obtain the corresponding inequality for the Carleson operator C, that is C : Lp(Mp+1w) → Lp(w) for any 1 < p < ∞ and any weight function w, with bound independent of w. We also provide a maximalmultiplier weighted theorem, a vector-valued extension, and more general two-weighted inequalities. Our proof builds on a recent work of Di Plinio and Lerner combined with some results on Orlicz spaces developed by Pérez.
2018-01-01T00:00:00ZWalter Rudin meets Elias M. SteinBakas, O.Ciccone, V.Wright, J.http://hdl.handle.net/20.500.11824/17342024-01-15T23:19:54Z2023-01-01T00:00:00ZWalter Rudin meets Elias M. Stein
Bakas, O.; Ciccone, V.; Wright, J.
Walter Rudin and Elias M. Stein were giants in the world of mathemat-
ics. They were loved and admired from students and researchers to teachers
and academics, both young and old. They touched many of us through their
inspiring books at the undergraduate and postgraduate level. Although they
were leading researchers in both harmonic analysis and several complex vari-
ables, we are not aware whether they interacted and discussed mathematics. In
this article, Rudin and Stein meet mathematically through a reformulation of
the beautiful theory of Fourier series with gaps that Rudin developed in the
1950s as an equivalent Fourier restriction problem from the 1970s, a problem
Stein proposed and which remains a fundamental, central problem in Euclidean
harmonic analysis today.
Walter Rudin was born in Vienna on 2 May, 1921 and emigrated to the US
in 1945, completing his PhD at Duke University in 1949. While a C. L. E.
Moore Instructor at MIT in the early 1950s, Walter was asked to teach a real
analysis course but he could not find a textbook that he liked so he decided to
write Principles of Mathematical Analysis which despite its age, has remained
the paragon of high quality. After a stint of teaching at the University of
Rochester, he took up a position at the University of Wisconsin, Madison in
1959 where he remained until his retirement as Vilas Professor in 1991. He died
at his home in Madison on 20 May, 2010.
Elias M. Stein (known to friends and colleagues as Eli) was born in Antwerp
on 13 January, 1931 and emigrated with his family to the US in 1941, settling in
New York where Eli attended high school. He went to the University of Chicago,
received his PhD in 1955, and then went to MIT as a C.L.E. Moore Instructor
before Antoni Zygmund told Eli “it’s time to return to Chicago.” In 1963, Stein
moved to Princeton University as a full professor where he remained until he
died on 23 December, 2018.
Between 2003 and 2011, Eli expanded the presentation of Walter’s Principles
and published a series of four books aimed at advanced undergraduates. This
series is quickly becoming an important part of any young analyst’s education.However the majority of books written by Rudin and Stein are postgraduate
textbooks and research monographs (too many to list here), mainly in the areas
of harmonic analysis and several complex variables where both men were central
figures.
In this article, these two luminaries meet in the world of mathematical anal-
ysis. We look back at some important work Rudin did in the 1950s and recast
it in terms of a far-reaching problem from the 1970s that Stein gave us.
2023-01-01T00:00:00Z