Young's inequality for convolutions

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Problem 18.2 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 18.1 "Convolution of functions in $L^1(\mathbb{R})$"

(a)(b)

Prove that if $f, g \in L_{1}(\mathbb {R})$ , then their convolution has the following properties:

(a)

$f * g \in L_{1}(\mathbb {R})$ and $\left\| f * g\right\|_{L_{1}(\mathbb {R})} \leqslant \left\| f\right\|_{L_{1}(\mathbb {R})} \cdot \left\| g\right\|_{L_{1}(\mathbb {R})}$ ;

(b)

$f * g = g * f$ .

Problem X.6.4 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section X.6 "The Fourier Transform and Differentiation"

If $f \in L^p(\mathbb {R})$ , $1 \leq p \leq \infty$ , and $g \in L^1(\mathbb {R})$ , show that $f * g \in L^p(\mathbb {R})$ and $\left\| f * g\right\|_p \leq \left\| f\right\|_p \left\| g\right\|_1$ .

Problem 13.7 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

Prove Theorem 13.8 for the cases $p = 1$ and $p = \infty$ .

Remark.

Theorem 13.8 (Young's Inequality): Fix $1 \leq p \leq \infty$ . If $f \in L^{p}(\mathbf{T})$ and $g \in L^{1}(\mathbf{T})$ then $f * g \in L^{p}(\mathbf{T})$ , and we have $\left\| f * g\right\|_{L^{p}} \leq \left\| f\right\|_{L^{p}}\left\| g\right\|_{L^{1}}$ .

Definition 13.6: The convolution of measurable 1-periodic functions $f$ and $g$ is the function $f * g$ given by $(f * g)(x) = \int_{0}^{1} f(y) g(x-y)\, d y$ , whenever this integral is well defined.

Problem 13.8 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

Prove Theorem 13.9.

Remark.

Theorem 13.9 (Young's Inequality): Fix $1 \leq p \leq \infty$ . If $f \in \ell^{p}(\mathbb {Z})$ and $g \in \ell^{1}(\mathbb {Z})$ then $f * g \in \ell^{p}(\mathbb {Z})$ , and we have $\left\| f * g\right\|_{\ell^{p}} \leq \left\| f\right\|_{\ell^{p}}\left\| g\right\|_{\ell^{1}}$ .

Problem 215 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection II.3.3 "Properties of the Lebesgue Integral"

(a)(b)

The convolution of functions $f_1$ and $f_2$ on the line is defined to be the function $f$ given by the formula

f(x)=\int _{-\infty }^{\infty }f_1(t)f_2(x-t)\, dt.

(a)

Prove that if $f_1$ and $f_2$ belong to $L^{1}\left(\mathbb {R}\right)$ , then the integrand is integrable for almost all $x$ , and that the convolution $f$ also belongs to $L^{1}\left(\mathbb {R}\right)$ .

(b)

Problem 522 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.1 "Convolutions of Test Functions"

Suppose that the numbers $p\geq 1$ , $q\geq 1$ , $r\geq 1$ are related by the equation $1/p+1/q-1/r=1$ . Prove that

L^{p}\left({G,\mu }\right)*L^{q}\left({G,\mu }\right)\subset L^{r}\left({G,\mu }\right).

Problem 2.8.4 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.8 "Integration on Product Spaces"

(a)(b)(c)(d)

Suppose $1 \leq p \leq \infty , f \in L^{1}\left(\mathbb {R}^{1}\right)$ , and $g \in L^{p}\left(\mathbb {R}^{1}\right)$ .

(a)

Imitate the proof of Theorem 8.14 to show that the integral defining $(f * g)(x)$ exists for almost all $x$ , that $f * g \in L^{p}\left(\mathbb {R}^{1}\right)$ , and that

\left\| f * g\right\| _{p} \leq \left\| f\right\| _{1}\left\| g\right\| _{p}

(b)

Show that equality can hold in (a) if $p = 1$ and if $p = \infty$ , and find the conditions under which this happens.

(c)

Assume $1 < p < \infty$ , and equality holds in (a). Show that then either $f = 0$ a.e. or $g = 0$ a.e.

(d)

Assume $1 \leq p \leq \infty , \epsilon > 0$ , and show that there exist $f \in L^{1}\left(\mathbb {R}^{1}\right)$ and $g \in L^{p}\left(\mathbb {R}^{1}\right)$ such that

\left\| f * g\right\| _{p} > (1-\epsilon )\left\| f\right\| _{1}\left\| g\right\| _{p}

Problem 10.5.9 from Vestrup E.M. "The Theory of Measures and Integration" [Ves03]More problems in Section 10.5 "Convolutions"

(a)(b)(c)(d)

Claim 6 (Young's Theorem): Let $1 < p < +\infty$ , let $f \in L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ with $g \in L^{p}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ . Then $f * g \in L^{p}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ , $g * f = f * g$ $\bar{\lambda }$ -a.e. on $\mathbb {R}$ , and $\left\| f * g\right\|_{p} = \left\| g * f\right\|_{p} \leq \left\| f\right\|_{1} \left\| g\right\|_{p}$ .

This exercise generalizes Young's Theorem and requires the result of Exercise 3 in Section 8.1. Show the following: Let $1 \leq p, q, r \leq +\infty$ be such that $\frac{1}{r} = \frac{1}{p} + \frac{1}{q} - 1$ . Let $f \in L^{p}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ and $g \in L^{q}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ . Then $f * g$ exists $\bar{\lambda }$ -a.e. on $\mathbb {R}$ and is in $L^{r}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ . Also, $\left\| f * g\right\|_{r} \leq \left\| f\right\|_{p} \left\| g\right\|_{q}$ .

(a)

If $p = +\infty$ , then $q = 1$ and $r = +\infty$ , and the claim follows easily in this case. The same comment applies to the case where $q = +\infty$ [and hence $p = 1$ and $r = +\infty$ ].

(b)

If $r = +\infty$ , then $p$ and $q$ are conjugate exponents, and the claim follows by Hölder's Inequality.

(c)

Assume that $1 \leq r < +\infty$ now. This forces $1 \leq p, q < +\infty$ . Show that it is sufficient to prove the claim when both $f, g \geq 0$ and $\left\| f\right\|_{p} = \left\| g\right\|_{q} = 1$ .

(d)

Prove the claim in (c).

Remark.

Claim 3: Let $f, g \in L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ . Then $f * g \in L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ and $\left\| f * g\right\|_{1} \leq \left\| f\right\|_{1} \left\| g\right\|_{1}$ .

Since $f, g$ are nonnegative and $\mathcal{L} / \mathcal{B}^{*}$ -measurable, we may meaningfully discuss $\int_{\mathbb {R}} f(x - y) g(y) \, d\bar{\lambda }(y) = f * g(x)$ for each $x \in \mathbb {R}$ . Show that $\frac{1}{r} + \frac{1}{q^{\prime }} + \frac{1}{p^{\prime }} = 1$ , and use Exercise 3 in Section 8.1 to obtain

\begin{aligned} f * g(x) & = \int _{\mathbb {R}} f(x - y) g(y) \, d\bar{\lambda }(y) \\ & = \int _{\mathbb {R}} f(x - y)^{\frac{p}{r}} g(y)^{\frac{q}{r}} f(x - y)^{1 - \frac{p}{r}} g(y)^{1 - \frac{q}{r}} \, d\bar{\lambda }(y) \\ & \leq \left[\int _{\mathbb {R}} f(x - y)^{p} g(y)^{q} \, d\bar{\lambda }(y)\right]^{1 / r} \left[\int _{\mathbb {R}} f(x - y)^{q^{\prime } \left(1 - \frac{p}{r}\right)} \, d\bar{\lambda }(y)\right]^{1 / q^{\prime }} \\ & \quad \times \left[\int _{\mathbb {R}} g(y)^{p^{\prime } \left(1 - \frac{q}{r}\right)} \, d\bar{\lambda }(y)\right]^{1 / p^{\prime }} \\ & = \left[\int _{\mathbb {R}} f(x - y)^{p} g(y)^{q} \, d\bar{\lambda }(y)\right]^{1 / r} \times 1 \times 1 \end{aligned}

for each $x \in \mathbb {R}$ . This shows that $\left(f * g\right)^{r} \leq f^{p} * g^{q} \in L^{1}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ by Claim 3. It follows that $f * g \in L^{r}\left(\mathbb {R}, \mathcal{L}, \bar{\lambda }\right)$ , and Claim 3 applied to $f^{p}$ and $g^{q}$ gives

\left\| f * g\right\| _{r}^{r} = \int _{\mathbb {R}} f * g(x)^{r} \, d\bar{\lambda }(x) \leq \int _{\mathbb {R}} f^{p} * g^{q}(x) \, d\bar{\lambda }(x) \leq \left\| f^{p}\right\| _{1} \left\| g^{q}\right\| _{1} = \left\| f\right\| _{p}^{p} \left\| g\right\| _{q}^{q} = 1 .

Problem 5.15 from Athreya K.B., Lahiri S.N. "Measure Theory and Probability Theory" [AtLa06]More problems in Chapter 5 "Product Measures, Convolutions, and Transforms"

(a)(b)

Let $f \in L^{1}(\mathbb {R}), g \in L^{p}(\mathbb {R}), 1 \leq p \leq \infty$ .

(a)

Show that if $1 \leq p < \infty$ , then for all $x$ in $\mathbb {R}$

\left|\int \right| f(u) g(x-u)|d u|^{p} \leq \left(\int |f| d m\right)^{p-1}\left(\int |g(x-u)|^{p}|f(u)| d u\right)

and hence that

(f * g)(x) \equiv \int f(u) g(x-u) d u

is well defined a.e. $(m)$ and

\left\| f * g\right\| _{p} \leq \left\| f\right\| _{1}\left\| g\right\| _{p}

with " = " holding iff either $f = 0$ a.e. or $g = 0$ a.e.

(b)

Show that if $p = \infty$ then

\left\| f * g\right\| _{\infty } \leq \left\| f\right\| _{1}\left\| g\right\| _{\infty }

and " $=$ " can hold for some nonzero $f$ and $g$ . (Hint for (a): Use Jensen's inequality with probability measure $d \mu = \frac{\left|f\right| d m}{\left\| f\right\|_{1}}$ if $\left\| f\right\|_{1} > 0$ .)

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[Ves03] Vestrup E.M. The Theory of Measures and Integration. Wiley-Interscience, 2003.

[AtLa06] Athreya K.B., Lahiri S.N. Measure Theory and Probability Theory. Springer, 2006.