Distributions and Fourier multipliers

A tempered distribution is a continuous linear functional on Schwartz space, using bilinear rather than conjugate-linear test pairing here. Define its Fourier transform by $\langle\widehat T,\phi\rangle=\langle T,\widehat\phi\rangle$ with the Euclidean negative-exponent convention. The Schwartz transform's continuity makes this another tempered distribution. For an ordinary integrable function embedded as a distribution, Fubini verifies agreement with its usual transform. This agreement is the reason for the dual definition, not an assumption that every distribution possesses an ordinary density.

The delta at zero evaluates a test function at zero, so its transform acts by integrating the test function and is the constant distribution one. Conversely the constant distribution transforms to a delta. Distributional differentiation is defined by moving a derivative onto the test function with a minus sign. Combining this rule with Fourier duality gives the familiar frequency multiplier for differentiation, now without requiring a classical derivative or an absolutely convergent transform integral.

For a bounded measurable symbol $m$, define $T_mf=\mathcal F^{-1}(m\widehat f)$ on L². Plancherel gives norm at most $\|m\|_\infty$, and testing on frequency sets where the symbol nearly reaches its essential supremum proves equality of operator norms. This definition uses multiplication of L² functions by a bounded measurable function. It should not be confused with multiplying an arbitrary distribution by an arbitrary measurable symbol, which is not generally defined.

A smooth symbol whose derivatives have at most polynomial growth does multiply Schwartz test functions continuously and therefore can multiply tempered distributions by duality. Polynomial symbols encode constant-coefficient differential operators, but they are usually unbounded on all of L². Their domains require enough frequency decay, as in Sobolev spaces. By contrast the heat multiplier at positive time is bounded and damps high frequencies. Writing both operations as multipliers clarifies their relationship while retaining the different domain and boundedness requirements.

For Euclidean dimension $d$, one convenient sufficient multiplier theorem assumes $m\in C^{d+2}(\mathbb R^d\setminus\{0\})$ and uniformly bounded quantities $|\xi|^{|\alpha|}|\partial^\alpha m(\xi)|$ for all multiindices of order at most $d+2$. Then the associated operator is bounded on Lᵖ for every $1<p<\infty$, with constants depending on these bounds, dimension, and p. This is a deliberately stronger-than-minimal form of the Hörmander–Mikhlin theorem; it supplies a safe usable hypothesis rather than the sharpest derivative count.

The theorem is not an endpoint assertion for L¹ or L∞, and merely bounded symbols need not be Lᵖ multipliers for other exponents. A proof decomposes the symbol into smooth frequency annuli, estimates their kernels, and uses singular-integral theory. That proof is deeper than the elementary Plancherel bound. When solving an equation by division in frequency, also check zeros of its symbol: division through a zero can introduce nonexistence, nonuniqueness, or compatibility conditions rather than a bounded inverse.