Minimizing sinc spectral lobes

Question

for some reasons I'd like to minimize the secondary lobes of the FFT transform of a rectangular signal (shwon in the upper-left side of picture below)

so I've used window function to do that (bottom-left). However, if I want to do it directly on the complex transform of the signal (bottom-right), the values of attenuation of secondary lobe aren't the same as the temporal case.

Having the test code below, I'd like to know what is wrong with it:

pass to the EDIT

     Fs = 1000; L = 2000; t = (0:L-1)*1/Fs;  
% the signal
y = zeros(1,L);  y(500:1500) = 1;     % rectangle signal

figure; 
subplot(2,2,1);
plot(Fs*t,y); title('Temporal signal'); xlabel('time (milliseconds)')

NFFT = 6*2^nextpow2(L);
Y1 = 2*abs(fft(y,NFFT)); Y1 = Y1/max(Y1);
f = 0:Fs/NFFT:Fs/2;

% the FFT of the signal
subplot(2,2,2);
semilogy(f, Y1(1:NFFT/2+1), '.-'); axis tight; grid on;
title('Signal''s spectrum'); xlabel('Frequency (Hz)'); ylabel('|Y(f)|');

w = window(@blackman, L);
y1 = y.*w';
Y2 = 2*abs(fft(y1,NFFT)); Y2 = Y2/max(Y2);

% the FFT of the windowed signal, in time
subplot(2,2,3);
semilogy(f, Y2(1:NFFT/2+1), '.-'); axis tight; grid on;
title('FFT windowing - temporal product'); xlabel('Frequency (Hz)'); ylabel('|Y(f)|')

% the FFT of the windowed signal, in complex
Y3 = 2*abs(fftshift(conv(Y1, fft(w,NFFT), 'same'))); Y3 = Y3/max(Y3);
subplot(2,2,4);
semilogy(f, Y3(1:NFFT/2+1), '.-');  axis tight; grid on;
title('FFT windowing - spectral convolution'); xlabel('Frequency (Hz)'); ylabel('|Y(f)|')

EDIT:
I've changed the code according to @Matt's remarks, but still there is a problem to the last graph (the noise on the convolutions of the FFTs)

NFFT = 6*2^nextpow2(L);
Y1 = fft(y,NFFT);
YY1 = 2*abs(Y1); YY1 = YY1/max(YY1);
f = 0:Fs/NFFT:Fs/2;

% the FFT of the signal
subplot(2,2,2);
semilogy(f, YY1(1:NFFT/2+1), '.-'); axis tight; grid on;
title('Signal''s spectrum'); xlabel('Frequency (Hz)'); ylabel('|Y(f)|');

w = window(@blackman, L);
y1 = y.*w';
YY2 = 2*abs(fft(y1,NFFT)); YY2 = YY2/max(YY2);

% the FFT of the windowed signal, in time
subplot(2,2,3);
semilogy(f, YY2(1:NFFT/2+1), '.-'); axis tight; grid on;
title('FFT windowing - temporal product'); xlabel('Frequency (Hz)'); ylabel('|Y(f)|')

% the FFT of the windowed signal, in complex
YY3 = 2*abs(cconv(Y1, fft(w,NFFT)', NFFT)); YY3 = YY3/max(YY3);
subplot(2,2,4);
semilogy(f, YY3(1:NFFT/2+1), '.-');  axis tight; grid on;
title('FFT windowing - spectral convolution'); xlabel('Frequency (Hz)'); ylabel('|Y(f)|')'

Answer 1

Multiplication of two time domain sequences corresponds to cyclic (circular) convolution of the DFTs of the two sequences, not to linear convolution as implemented in your code. So if $x[n]$ and $y[n]$ are two length $N$ time domain sequences, and $X[k]$ and $Y[k]$ are their respective DFTs, the following correspondence holds:

$$\text{DFT}_N\left\{x[n]\cdot y[n]\right\}[k]= \frac{1}{N}\sum_{m=0}^{N-1}X[m]Y[(k-m)_N],\quad 0\le k \le N-1$$

where

$$(k-m)_N=k-m+lN$$

with the integer $l$ chosen such that the result is in the range $[0,N-1]$.

Also take a look at this answer to a related question.

EDIT: You also can't use Y1 in the convolution because it is not the DFT but the magnitude of the DFT. You should circularly convolve fft(y) with fft(w).