$L^p$-Fourier multipliers with bounded powers

V. V. Lebedev ; A. M. Olevskii

Izvestiya. Mathematics , Tome 70 (2006) no. 3, pp. 549-585.

Résumé

We consider the space $M_p(\mathbb R^d)$ of $L^p$-Fourier multipliers and give a detailed proof of the following result announced by the authors in $\lbrack10\rbrack$: if $\varphi\colon\mathbb R^d\to \lbrack0, 2\pi\lbrack$ is a measurable function and $\|e^{in\varphi}\|_{M_p}=O(1)$, $n\in\mathbb Z$, for some $p\ne 2$, then the function $\varphi$ is linear in domains complementary to some closed set $E(\varphi)$ of Lebesgue measure zero, and the set of values of the gradient of $\varphi$ is finite. We also consider the question of which sets can appear as $E(\varphi)$. We study the behaviour of the norms of the exponential functions $e^{i\lambda\varphi}$ in the case when the frequency $\lambda$ tends to infinity along a sequence of real numbers. In particular, we construct a homeomorphism $\varphi$ of the line $\mathbb R$ which is non-linear on every interval and satisfies $\|e^{i2^n\varphi}\|_{M_p(\mathbb R)}=O(1)$, $n=0, 1, 2,\dots$, for all $p$, $1$.

Bibliographie

[1] Khermander L., Otsenki dlya operatorov, invariantnykh otnositelno sdviga, IL, M., 1962

[2] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR | Zbl

[3] De Leeuw K., “On $L^p$ multipliers”, Ann. Math., 81:2 (1965), 364–379 | DOI | MR | Zbl

[4] Edwards R. E., Gaudry G. I., Littlewood–Paley and multiplier theory, Springer-Verlag, Berlin–Heidelberg–N. Y., 1977 | MR | Zbl

[5] Olevskiǐ A., “Six lectures on translation-invariant operators and subspaces”, Rend. Instit. Mat. Univ. Trieste, XXXI:1 (2000), 203–233 | MR | Zbl

[6] Beurling A., Helson H., “Fourier transforms with bounded powers”, Math. Scand., 1 (1953), 120–126 | MR | Zbl

[7] Kakhan Zh.-P., Absolyutno skhodyaschiesya ryady Fure, Mir, M., 1976

[8] Kahane J.-P., “Quatre leçons sur les homéomorphismes du cercle et les séries de Fourier”, Topics in Modern Harmonic Analysis, V. II, Ist. Naz. Alta Mat. Francesco Severi, Roma, 1983, 955–990 | MR | Zbl

[9] Lebedev V., Olevskiǐ A., “$C^1$ changes of variable: Beurling–Helson type theorem and Hörmander conjecture on Fourier multipliers”, Geometric and Functional Analysis (GAFA), 4:2 (1994), 213–235 | DOI | MR | Zbl

[10] Lebedev V., Olevskiǐ A., “Bounded groups of translation invariant operators”, C. R. Acad. Sci. Paris, Série I, 322:2 (1996), 143–147 | MR | Zbl

[11] Lebedev V., Olevskiǐ A., “Idempotents of Fourier multiplier algebra”, Geometric and Functional Analysis (GAFA), 4:5 (1994), 539–544 | DOI | MR | Zbl

[12] Jodeit M., “Restrictions and extensions of Fourier multipliers”, Studia Mathematica, 34 (1970), 215–226 | MR | Zbl

[13] Katznelson Y., An Introduction to Harmonic Analysis, Dover, N. Y., 1976 | MR | Zbl

[14] Graham C. C., “Mappings that preserve Sidon sets in $\mathbb R$”, Arkiv för Matematik, 19 (1981), 217–221 | DOI | MR | Zbl

[15] Sjögren P., Sjölin P., “Littlewood–Paley decompositions and Fourier multipliers with singularities on certain sets”, Ann. Inst. Fourier, 31:1 (1981), 157–175 | MR | Zbl

[16] Rubio de Francia J. L., “A Littlewood–Paley inequality for arbitrary intervals”, Rev. Mat. Iberoam., 1:2 (1985), 1–14 | MR | Zbl

[17] Hare K. E., Klemes I., “On permutations of lacunary intervals”, Trans. Am. Math. Soc., 347:10 (1995), 4105–4127 | DOI | MR | Zbl

[18] Riss F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979 | MR