Question Number 218896 by Nicholas666 last updated on 17/Apr/25
![prove; ∣∫∫∫_([0,∞]^3 ) f((J_0 (x)J_0 (y)J_0 (z))/(1+x^2 y^2 z^2 ))∣≤C(∫∫∫_R_+ ^3 ∣f∣(1+x^2 y^2 z^2 )^2 )^(1/2)](https://www.tinkutara.com/question/Q218896.png)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{prove}}; \\ $$$$\:\mid\int\int\int_{\left[\mathrm{0},\infty\right]^{\mathrm{3}} } \boldsymbol{{f}}\frac{\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{x}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{y}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{z}}\right)}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} }\mid\leqslant\boldsymbol{{C}}\left(\int\int\int_{\mathbb{R}_{+} ^{\mathrm{3}} } \mid\boldsymbol{{f}}\mid\left(\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} \right)^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:\:\:\:\:\:\: \\ $$$$ \\ $$
Commented by Nicholas666 last updated on 17/Apr/25
![Information; • ∫∫∫_R_+ ^3 =∫_0 ^∞ ∫_(0 ) ^∞ ∫_0 ^∞ dxdydz • ∫∫∫_([0,1] ) = ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 dxdydz • f = is an integrable reel function or an integrable quadratic function on the appropriate domain • C= is a univeral positive constan that does not depend on f • J_n (t) is the first kind Bessel function of orden n](https://www.tinkutara.com/question/Q218897.png)
$$\:\:\:\:\boldsymbol{{Information}}; \\ $$$$\bullet\:\:\:\int\int\int_{\mathbb{R}_{+} ^{\mathrm{3}} } =\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}\:} ^{\infty} \int_{\mathrm{0}} ^{\infty} \boldsymbol{{dxdydz}} \\ $$$$\:\bullet\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]\:} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{{dxdydz}} \\ $$$$\bullet\:\:\:\boldsymbol{{f}}\:=\:\boldsymbol{{is}}\:\boldsymbol{{an}}\:\boldsymbol{{integrable}}\:\boldsymbol{{reel}}\:\boldsymbol{{function}}\:\boldsymbol{{or}}\:\boldsymbol{{an}} \\ $$$$\:\:\:\:\:\boldsymbol{{integrable}}\:\boldsymbol{{quadratic}}\:\boldsymbol{{function}}\:\boldsymbol{{on}}\:\boldsymbol{{the}}\:\boldsymbol{{appropriate}}\:\boldsymbol{{domain}}\:\:\:\:\:\:\: \\ $$$$\:\bullet\:{C}=\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{univeral}}\:\boldsymbol{{positive}}\:\boldsymbol{{constan}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{that}}\:\boldsymbol{{does}}\:\boldsymbol{{not}}\:\boldsymbol{{depend}}\:\boldsymbol{{on}}\:\boldsymbol{{f}}\: \\ $$$$\:\bullet\:\:\boldsymbol{{J}}_{\boldsymbol{{n}}} \left(\boldsymbol{{t}}\right)\:\boldsymbol{{is}}\:\boldsymbol{{the}}\:\boldsymbol{{first}}\:\boldsymbol{{kind}}\:\boldsymbol{{Bessel}}\:\boldsymbol{{function}}\:\boldsymbol{{of}}\:\boldsymbol{{orden}}\:\:\boldsymbol{{n}} \\ $$$$ \\ $$