I work in Fourier analysis , or more generally harmonic analysis, and applications . I am interested in singular integrals and
Calderón-Zygmund operators, function spaces,
Littlewood-Paley theory, discrete decompositions, and wavelets.
Applications of this theoretical research
include partial differential equations and signal analysis .
I am also involved in interdisciplinary work in the
spectral analysis of nanostructures in biological tissues.
In general terms,
Fourier analysis is a mathematical tool that permits the decomposition of
a function or signal into a combination
of oscillating waves of different frequencies
and amplitudes, very much in the same way that a prism separates
a beam of
light into a rainbow of colors of different wavelengths.
Fourier analysis and related
mathematical techniques decode information present in signals or
sets of data and provide a precise mechanism and quantitative way
to analyze variations, oscillations, and sudden changes in the data, as well as trends, patterns, and
symmetries.
Selected Recent Works
- Discrete Decompositions for Bilinear Operators
and Almost Diagonal Conditions (with L. Grafakos), Trans. Amer. Math. Soc. 354 (2002), no. 3, 1153-1176.
(Gzipped PostScript)
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On multilinear singular integrals of Calderón-Zygmund type (with L. Grafakos), Publ. Mat. (2002), 57-91.
- Multilinear Calderón-Zygmund Theory (with L. Grafakos), Adv. Math. 165
(2002), no. 1, 124-164.
(Gzipped PostScript) (PDF)
- Maximal Operator and Weighted Norm Inequalities for
Multilinear Singular Integrals (with L. Grafakos), Indiana Univ. Math. J. 51 (2002), 1261-1276
(Gzipped PostScript) (PDF)
- Decomposition of $\dot B^{0,1 _1}(Z)$ into special atoms (with S. Boza), Math. Nachr. 254/255 (2003), 3-10.
(Gzipped PostScript)
(PDF)
- Uniqueness in the inverse conductivity problem for conductivites with 3/2 derivatives in L^p , p > 2n (with R. Brown), J.
Fourier Anal. and Appl. 9 (2003), 563-574. (ps)
(PDF)
- Coherent light scattering of ultraviolet light by avian feather barbs (with R. Prum and S. Andersson), Auk 120 (2003), 163-170.
- Sharp maximal function estimates for
multilinear singular integrals, (with C. Pérez), Contemp. Math. 320 (2003), 323-331.
- Symbolic calculus for the transposes of
bilinear pseudodifferential operators (with A. Bényi), Comm. Partial Diff. Eq. 28 (2003), 1161-1181.
- Structural colouration of avian skin: convergent evolution of coherently scattering dermal collagen arrays (with R. Prum), Journal of Experimental Biology 206 (2003), 2409-2429. pdf
- A Fourier tool for the ananlysis of coherent light scattering by bio-optical nanostructures (with R. Prum), Integr. Comp. Biol. 43 (2003), 591-602.
- Almost orthogonality and a class of bounded pseudodifferential operators (with A. Bényi), Mathematics Research Letters 11 (2004), 1-11.
- Structural colouration of mammalian skin: convergent evolution of coherently scattering dermal collagen arrays (with R. Prum), Journal of Experimental Biology 207 (2004), 2157-2172. pdf
- Calderón-Zygmund operators on mixed Lebesgue
spaces and applications to null forms (with A. Stefanov), J. London Math. Soc. 70 (2004), 447-462.
- Blue integumentary structural colours in dragonflies (Odonata) are not produced by incoherent Tyndall scattering (with R. Prum and J. Cole), Journal of Experimental Biology 207 (2004), 3999-4009. pdf
- Análisis espectral de nanoestructuras en tejidos biológicos (with R. Prum),
Matematicalia 1 no. 2 (2005) http://www.matematicalia.net/.
- Anatomically Diverse Butterfly Scales Produce Structural Colors by Coherent Scattering (with R. Prum and T. Quinn), Journal of Experimental Biology 209 (2006), 748-765.
pdf
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Sobolev space estimates and symbolic calclus for bilinear pseudodifferential operators (with A. Bényi and A.R. Nahmod), J. Geom. Anal. 3 (2006), 431-454. http://www.mathjournals.org/jgan/2006-016-003/
- Modulation invariant bilinear T(1) theorem (with A. Bényi, C. Demeter, A.R. Nahmod, C.M. Thiele, and F. Villarroya), J. Anal. Math. 109 (2009), 279-352 http://arxiv.org/abs/0710.0973.
- Bilinear paraproducts revisited (with A. Bényi, D. Maldonado, and A.R. Nahmod) Math. Nachr., to appear.
- New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory (with A Lerner, S. Ombrosi, C. Pérez and R. Trujillo-González) Adv. Math. 220 (2009), no. 4, 1222--1264.
- On the Hormander classes of bilinear pseudodifferential operators (with A. Bényi, D. Maldonado and V. Naibo, J. Integral Eq. Oper. Theory, to appear http://arxiv.org/abs/0909.4734.
- Sharp weighted bounds for fractional integral operators (with M. Lacey, K. Moen, and C. Pérez), submitted http://arxiv.org/abs/0905.3839 .
Talk given at Seville 08
Support and Disclaimer
This research has been supported in part by the National Science Foundation under the following Grants: DEB-9318273, DMS-9696267, DMS-0070514, DBI-0078376, DMS-0112375, DMS-0400423, DMS-0800492.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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